Study of Smarandache Functions and Sequences

关于Smarandache问题 研究的新进展 郭晓艳 西北大学数学系 袁 霞 西北大学数学系 High American Press 2010 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. Zeeb Road P.O. Box 1346, Ann Arbor MI 48106-1346, USA Tel.: 1-800-521-0600 (Customer Service) http://wwwlib.umi.com/bod/basic Peer Reviewers: Wenpeng Zhang, Department of Mathematics, Northwest University, Xi’an, Shannxi , P.R.China. Wenguang Zhai, Department of Mathematics, Shangdong Teachers’ University, Jinan, Shandong , P.R.China. Guodong Liu, Department of Mathematics, Huizhou University, Huizhou,Guangdong, P.R.China. Copyright 2010 by High Am. Press, translators, editors, and authors for their papers Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm ISBN: 978-1-59973-096-7 Standard Address Number : 297-5092 Printed in the United States of America ó êØ´˜€ïÄê5Æ, AO´ê5Ÿ‰Æ. l§)ƒF å, Ò±Šó{', Vg˜ß, Øä²(kOuÙ‰Æ. êÆ fpdQ²`L “êÆ´‰Æå, êØK´êÆå”. êØ ´˜€PêÆƉ, P§Œ±Jˆž“<‚(-P¯! , êØqé ”, ”·‚y3, {(½êNõ{ü5Ÿ. Ǒ,kNõPêدK®²)û, ´qkõ#¯KØä Ñy. duNõêدKïāªþŒ=zǑ, êؼê5?Ø, Ï déêؼêïʆ´êØ¥˜‡ÄǑ´­‡ïđK. 1993 , 35Only Problem, Not Solutions!6˜Ö¥, {7ÛêZæͶ êØ;[ F. Smarandache ÇJÑ 105 ‡'uAÏê!Žâ¼ê ™)ûêƯK9ߎ. ‘Xù ¯KJÑ, NõÆöéd?1 \ïÄ, ¿¼ Øäk­‡nØdŠïĤJ. Ö´Šö3ÜŒÆÖÆ Ïm, Šâ“Ü©+ÇïÆ, ò8 ISÆö'u Smarandache ¯KïÄÜ©¤J®?¤þ, ÙÌ ‡83u•Öö0 'u Smarandache ¯K˜ #ïĤJ, ̇) Smarandache ¼êk.5O!þŠO, AÏê, AÏ ¼ê§)˜X¯K. F"k,ÖöŒ±éù (ØÚ#¯ K?1ïÄ, l mÿÖöÀ, ÚÚ-uÖöéù +ïÄ, .  , é “Ü©+Çå|±Ú9œy, ["Ö¿J ÑNõB¿„—±¿! ?ö 2010 12  I 'uSmarandache¯KïÄ#? 8¹ 1˜Ù 'u Smarandache ¼ê 1.1 'u Smarandache ¼êe.O . . . . . . . . . . Úó9(Ø . . . . . . . . . . . . . . . . . . 1.1.1 1.1.2 ½n 1.1 y² . . . . . . . . . . . . . . . . 1.2 Smarandache ¼ê3ê a + b þe.O . . . . 1.2.1 Úó9ïĵ . . . . . . . . . . . . . . . . 1.2.2 ½n 1.2 y² . . . . . . . . . . . . . . . . 1.3 Smarandahce ¼ê3¤êêþe.O . . . . . . 1.3.1 ¤êê†Ì‡(Ø . . . . . . . . . . . . . . 1.3.2 ½n 1.3 y² . . . . . . . . . . . . . . . . 1.4 Smarandache ¼ê3 £þe.O . . . . . . 1.4.1  £0 . . . . . . . . . . . . . . . . . ½nÚü‡íØy² . . . . . . . . . . . . . 1.4.2 1Ù 'u Smarandache LCM ¼ê˜ ¯K Úó . . . . . . . . . . . . . . . . . . . . . . . 2.1 2.2 'u Smarandache LCM ¼ê9Ùéó¼ê . . . . . . 2.3 Smarandache LCM ¼êéó¼ê†ƒÏfþ Š........................ 2.4 ˜‡¹ Smarandache LCM ¼êéó¼ê§ . . 2.5 Smarandache ¼ê† Smarandache LCM ¼ê·ÜþŠ 2.6 ˜‡¹ Smarandache ¼ê† Smarandache LCM ¼ê § . . . . . . . . . . . . . . . . . . . . . . 1nÙ 'u Smarandache Ú¼ê˜ ¯K 3.1 'u Smarandache Ú¼êþŠ . . . . . . . . . . . 3.1.1 Úó9(Ø . . . . . . . . . . . . . . . . . . 3.1.2 ½n 3.1 9½n 3.2 y² . . . . . . . . . . . ˜a¹ Smarandache Ú¼ê S(n, k)  Dirichlet ?ê . 3.2 3.2.1 Dirichlet ?ê†Ì‡(Ø . . . . . . . . . . . . 3.2.2 A‡½ny² . . . . . . . . . . . . . . . . p II p 1 1 1 3 5 5 5 9 9 10 13 13 14 18 18 18 22 24 30 33 37 37 37 38 42 42 43 8¹ 3.3 ˜a¹ Smarandache Ú¼ê AS(n, k)  Dirichlet ?ê 46 3.3.1 ̇(Ø . . . . . . . . . . . . . . . . . . . 46 ½ny² . . . . . . . . . . . . . . . . . . 48 3.3.2 3.4 'u Smarandache ˜ÚþŠ . . . . . . . . . . . . 51 3.4.1 ïĵ9̇(Ø . . . . . . . . . . . . . . 51 3.4.2 ½n 3.13 9½n 3.14 y² . . . . . . . . . . 53 1oÙ 'uŒ\¼ê˜ ¯K 56 4.1 ˜‡#Œ\¼ê† Smarandache ê . . . . . . . . 56 4.1.1 Úó9(Ø . . . . . . . . . . . . . . . . . . 56 4.1.2 ü‡{üÚn . . . . . . . . . . . . . . . . 57 ½n 4.1 9½n 4.2 y² . . . . . . . . . . . 59 4.1.3 4.2 'uŒ\¼êþŠ . . . . . . . . . . . . . . . 61 4.2.1 ̇(Ø . . . . . . . . . . . . . . . . . . . 61 4.2.2 ½n 4.3 y² . . . . . . . . . . . . . . . . 61 1ÊÙ 'u Smarandache ê9Ùk'¯K 66 5.1 Smarandache ²ê SP (n) Ú IP (n) þŠ . . . 66 5.2 Smarandache 3n-digital ê . . . . . . . . . . . . . 69 18Ù ˜ ¹ Smarandache ¼ê§ 76 ¹– Smarandache ¼êÚ Smarandache LCM ¼ê 6.1 § . . . . . . . . . . . . . . . . . . . . . . . 76 6.2 ˜‡¹ Smarandache ¼ê†– Smarandache ¼ê § . . . . . . . . . . . . . . . . . . . . . . . . 85 6.3 'u Smarandache ¼êü‡ßŽ . . . . . . . . . . 88 ˜‡¹¼ê S (n) § . . . . . . . . . . . . . 93 6.4 6.5 'u Smarandache ¯K˜‡í2 . . . . . . . . . . 100 1ÔÙ Smarandache ¼êƒ'¯K 105 7.1 Smarandache ¼ê·ÜþŠ¯K . . . . . . . . . . 105 7.2 'u²Öê SSC(n) ü‡¯K . . . . . . . . . . 109 7.3 'u Smarandache ˜¼ê˜‡þŠ¯K . . . . . . . 113 7.4 'u Smarandache {ü¼ê . . . . . . . . . . . . . 120 7.5 Smarandache k gÖê¼ê . . . . . . . . . . . . . 123 7.6 ˜‡¹ Gauss ¼ê§9Ù¢ê) . . . . . . . . 127 k III 7.7 ë©z IV 'uSmarandache¯KïÄ#? n ?›¥š"êi겐ڼêþŠ . . . . . . . . 131 135 1˜Ù 'u Smarandache ¼ê 1˜Ù 'u Smarandache ¼ê êØ¥¤¹˜‡­‡SNÒ´ïÄêؼꈫ5Ÿ, Ͷ Smarandache ¼ê S(n) ´­‡êؼꃘ, éuù˜¼ êéõÆö®²‰ ïÄÚ&¢, ¿ ˜X­‡(J, ù nØ ¤JéêØuÑk­Œ¿Â. C 5, 'u Smarandache ¼êk. 5O¯K¤Ǒ Smarandache ˜‡#,‘K, éõÆöéù˜‘K‰ Ǒ&¢, Ùò0 CÏISÆö‚'u Smarandache ¼êk .5O¯K¤ŠÑ#¤J. 1.1 1.1.1 'u Smarandache ¼êe.O Úó9(Ø ½Â 1.1. éu?¿ê n, Ͷ Smarandache ¼ê S(n) ½Â Ǒê m  n | m!. =Ò´ S(n) = min{m : m ∈ N, n|m!}. l S(n) ½ÂéN´íÑXJ n = p p · · · p L« n IO ©)ª, o S(n) = max {S(p )}. dd·‚ǑØJOŽÑ S(1) = 1, α1 α2 1 2 1≤i≤r αr r αi i S(2) = 2, S(3) = 3, S(4) = 4, S(5) = 5, S(6) = 3, S(7) = 7, S(8) = 4, S(9) = 6, S(10) = 5, S(11) = 11, S(12) = 4, S(13) = 13, S(14) = 7, S(15) = 5, S(16) = 6, S(17) = 17, S(18) = 6, S(19) = 19, S(20) = 5, · · · . S(n) , . S(n) , , , [2-6]. , [2] w,¼ê QØ´4O¼ê ǑØ´4~¼ê 'u ?˜Ú 5Ÿ NõÆöǑ?1 ïÄ ¼ Øk(J ë©z ~X ºæ² ¥ïÄ § S (m1 + m2 + · · · + mk ) = k X S (mi ) i=1 Œ)5, |^)ÛêإͶnƒê½ny² é?¿ê k ≥ 3, T§k ¡õ|ê) (m , m , · · · , m ). 1 2 k 1 'uSmarandache¯KïÄ#? Mó¸ [3] ïÄ S(n) Š©Ù¯K, y² ìCúª
3 2ζ 32 x 2 2 +O (S(n) − P (n)) = 3 ln x n≤x X 3 x2 ln2 x ! , Ù¥ P (n) L« n ŒƒÏf, ζ(s) L« Riemann zeta- ¼ê. WjuÇ3©z [4] ¥ïÄ S 2 (2 − 1)
e.O¯K, ¿‰Ñ Oª:
p−1 p S 2p−1 (2p − 1) ≥ 2p + 1, Ù¥ p Ǒ?¿Ûƒê. €ïw [5] ¥U? ©z [4] (Ø, ‰Ñ re.O. =Ò ´y² é?¿ƒê p ≥ 7, ·‚k
S 2p−1 (2p − 1) ≥ 6p + 1. €ïw [6] ¥„ïÄ S (2 ê p ≥ 7, ÓŒOª p + 1) e.O¯K, y² é?¿ƒ S (2p + 1) ≥ 6p + 1. ±þ©z¥¤9ê 2 (2 − 1) kX­‡êص, ¯¢ þê M = 2 − 1 ¡ǑrÜZê. rÜZQßÿé¤kƒê p, M Ǒ ƒê. , ù˜ßÿ 5y´†Ø, ÏǑ M = 2 − 1 = 23 × 89 ´‡Üê. ê 2 (2 − 1) †˜‡PêØJK — óê— ƒƒ'. ¤¢ n ´˜‡괍 n ¤kÏêƒÚu 2n. ~ X n = 6 ´˜‡ê, ÏǑ 12 = 2 × 6 = 1 + 2 + 3 + 6. <‚®y² ˜‡óê n ´ê = n = 2 (2 − 1), Ù¥ 2 − 1 Ǒƒê. ´ Ä3Ûê–8´˜‡™)ûêØJK. k'SNŒë©z [8] 9 [9]. éu Smarandache ¼ê3Ù§êþe.O, ˜ ÆöǑ?1 ïÄ, ~X, <a [7] ?Ø Smarandache ¼ê3¤êêþe. O¯K, y² é?¿ê n ≥ 3 kOª: p−1 p p p p 11 p−1 p p−1 2 11 p
n S(Fn ) = S 22 + 1 ≥ 8 · 2n + 1, p 1˜Ù 'u Smarandache ¼ê Ù¥ F = 2 + 1 ǑͶ¤êê. C, É©z [5]![6] 9 [7] éu§§X¶ [10] ïÄ k'¯K, ¼ re.O. äN/`ǑÒ´y² e¡: ½n 1.1. éu?¿ƒê p ≥ 17, ·‚kOª 2n n (A). S (2p − 1) ≥ 10p + 1; (B). S (2p + 1) ≥ 10p + 1. w,½n 1.1 ¥e.O`u©z [4]![5]![6] 9 [7] ¥(Ø, §y²L§äkE|5. ½n 1.1 y² 1.1.2 ù!·‚|^{9|ÜE|†‰Ñ½n 1.1 y². ·‚y²½n 1.1 ¥ (A) ª, ÓnŒíѽn 1.1 ¥ (B) ª. d Smarandache ¼ê5Ÿéu?¿ƒê p | n, ·‚k S(n) ≥ p p | S (p ) é¤kê α ¤á. y3, éu?¿ƒê p ≥ 17,  q Ǒ (2 − 1) ?˜ƒÏf, w, q ≥ 5. u´d S(n) 5Ÿ α p S (2p − 1) ≥ q. (1-1) q = 2kp + 1, k = 1, 2, 3, · · · · · · . (1-2) qdu q | 2 − 1, ¤± 2 ≡ 1 (mod q). Ïd p ´ 2  q I. ¤± d©z [8] 9 [9] ¥I5Ÿ p | φ(q) = q − 1, ½ö q = mp + 1. d u q Ǒۃê, ¤± m ˜½Ǒóê, ÏdŒ p p w, 2 − 1 ،U´˜‡²ê. ÄKk 2 − 1 = u , ½ö 2 = u + 1, ddíÑ 0 ≡ 2 ≡ u + 1 ≡ 2 (mod 4), gñ. u´ 2 − 1 ke Ê«ŒU: (a). 2 − 1 Ǒƒê, dž5¿ p ≥ 17, ·‚k S (2 − 1) ≥ 2 − 1 ≥ 10p + 1. (b). 2 − 1 TǑ˜‡ƒê q  m g˜, m ≥ 3. du 2 − 1 ، UǑ², ¤± m = 3, 5, · · · . e m ≥ 5, Kdž(Ü (1-1) 9 (1-2) ªk p p 2 p p 2 2 p p p p p p S (2p − 1) ≥ S(q m ) ≥ mq > 5(2p + 1) > 10p + 1. 3 'uSmarandache¯KïÄ#? e m = 3, K q = 2kp + 1 k ≥ 2 žEk S (2p − 1) ≥ S(q 3 ) ≥ 3q > 3(4p + 1) > 10p + 1. w, 2 − 1 6= (2p + 1) , ÏǑ p ≥ 17 žª 2 − 1 = (2p + 1) Ø ŒU¤á, ÏǑ 2 − 1 > (2p + 1) , XJ p ≥ 17. (c). 2 − 1 –¹ko‡ØÓƒÏf. džd (1-2) ªŒ–k ˜‡ƒê÷v q = 2kp + 1 k ≥ 5, ÏǑ 2p + 1 Ú 4p + 1 ،UӞ Ǒƒê. džÒk S (2 − 1) ≥ q ≥ 10p + 1. (d). 2 − 1 T¹kn‡ØÓƒÏf, XJÙ¥–k˜‡ƒÏ f÷v q = 2kp + 1 k ≥ 5, oÒk S (2 − 1) ≥ q ≥ 10p + 1. XJ¤ kƒÏf¥ k ≤ 4, K5¿ 2p + 1 Ú 4p + 1 ،UӞǑƒê, 4p + 1 Ú 8p+1 ،UӞǑƒê, ¤±Œ 2 −1 = (2p+1) ·(6p+1) ·(8p+1) . dž β ≥ 2 ½ö γ ≥ 2 ½ö α ≥ 5 ž½nw,¤á. ¤±Ø”˜„5 Œb½ 2 −1 = (2p+1) ·(6p+1)·(8p+1), 1 ≤ α ≤ 4. ù«œ¹Ǒ´ØŒ U. ÏǑXJ 2 −1 = (2p+1) ·(6p+1)·(8p+1), Kdg{5ŸŒ  2 ´ƒê 2p+1 9 6p+1 g{. ,  p ≡ 3 (mod 4) ž,  p =  = (−1) = (−1) = (−1) = −1, ù 4k + 3, dž † 2 ´ƒê 6p + 1 g{gñ.  p ≡ 1 (mod 4) ž,  p = 4k + 1,   dž = (−1) = (−1) = −1§ù† 2 ´ = (−1) ƒê 2p + 1 g{gñ. ¤± 2 − 1 T¹kn‡ØÓƒÏf ž, ˜½k S (2 − 1) ≥ 10p + 1. (e). 2 − 1 T¹kü‡ØÓƒÏf. dž5¿ (1-2) ª± 9 (d) ¥y²L§Œ 2 − 1 ،UӞ¹kƒÏf 2p + 1 9 6p + 1. Ӟ 2 − 1 Ǒ،UӞ¹kƒÏf 2p + 1 Ú 4p + 1, ÏǑƒê p > 3 ž, ü‡ê 2p + 1 9 4p + 1 ¥–k˜‡ 3 Ø, Ïd§‚ØŒU ӞǑƒê. ¤±d (1-2) ª 2 − 1 T¹kü‡ØÓƒÏfž Œ: 2 − 1 = (2p + 1) · (8p + 1) ½ö 2 − 1 = (4p + 1) · (6p + 1) , ÏǑ 4p + 1 Ú 8p + 1 ،UӞǑƒê, Ù¥–k˜‡ 3 Ø. w, β ≥ 2 ½ö α ≥ 5 žk S (2 − 1) ≥ β · (6p + 1) ≥ 10p + 1 ½ö S (2 − 1) ≥ α · (2p + 1) ≥ 10p + 1.  β = 1, 1 ≤ α ≤ 4 ž k 2 − 1 = (2p + 1) · (8p + 1) ½ö 2 − 1 = (4p + 1) · (6p + 1). e 2 −1 = (2p+1) ·(8p+1), w, α 6= 4. ÄKd 2 −1 = (2p+1) ·(8p+1) áǑíÑÓ{ª: 2 − 1 ≡ −1 ≡ (2p + 1) · (8p + 1) ≡ 1 (mod 8), g ñ. 3 2 − 1 = (4p + 1) · (6p + 1) ¤áž, E,k S (2 − 1) ≥ p 3 p p 3 3 p p p p p p α β γ α p α (6p+1)2 −1 8 2 6p+1 (2p+1)2 −1 8 2 2p+1 3p(3p+1) 2 p(p+1) 2 6k+5 2k+1 p p p p p p p α β p α β p p p α p p α p p p 4 α 4 4 3 p 1˜Ù 'u Smarandache ¼ê 3 · (4p + 1) > 10p + 1. ¤±Ø” 1 ≤ α ≤ 3. dž p ≥ 17 ž,  ª 2 − 1 = (2p + 1) · (8p + 1) ½ö 2 − 1 = (4p + 1) · (6p + 1) ،U ¤á, ÏǑ 2 − 1 > (2p + 1) · (8p + 1) 9 2 − 1 > (4p + 1) · (6p + 1). n܈«ŒU·‚ØJíÑ 2 − 1 T¹kü‡ØӃÏfž, S (2 − 1) ≥ 10p + 1. (ܱþÊ«œ¹·‚áǑ¤½n 1.1 ¥ (A) ªy². aq/, ·‚Œ±íѽn 1.1 ¥ (B) ª. p 3 p p 2 3 p 2 p p 1.2 1.2.1 Smarandache O ¼ê3ê a + b þe. p p Úó9ïĵ ŠǑþ!¯Kí2Úò, <‚g,¬Ž Smarandache ¼êé ?¿g,êe.O. , ù´˜‡›©(J¯K, ÏǑ n = p Ǒ ƒêž, S(p) = p;  n = p α ≤ p ž, S (p ) = α · p. ¤± S(n) Š©ÙéØþ!. !ò0 o®Ú†Ü©+Ç3ù¡ï Ä(J, ǑÒ´ŠǑ©z [6] 9 [7] 5º, |^9|ܐ{ïÄ Smarandache ¼ê S(n) 3AÏê a + b þe.O¯K, ¿  ˜‡˜„5(Ø. äN/`ǑÒ´y² e¡: ½n 1.2.  p ≥ 17 Ǒƒê, Ké?¿ØÓê a 9 b, ·‚ kOª α α p p S (ap + bp ) ≥ 8p + 1. w,ù‡½n¥e.´©z [5] 9 [6] í2Úò, AO a = 2, b = 1 ž, ·‚áǑíÑOª S (2 + 1) ≥ 8p + 1. Ïd·‚Ǒ| ^ù!{Œ±U?©z [4]![5] ¥e.. p 1.2.2 ½n 1.2 y² ù!·‚|^{‰Ñ½n 1.2 y². ǑQãB, ·‚Äk ‰Ñe¡: 5 Ún 1.2.1. ·‚k y²:  K (h, k) = 1 2 'uSmarandache¯KïÄ#?  p Ǒۃê, Ké?¿pƒê a 9 b   ap + bp , a+b a+b ap + bp , a+b a+b p p   =1 ½ö p. = d, a + b = dh, p p d hk = a + b = a + (dh − a) = = pdhap−1 + p−2 X a+b 6= 0, p−1 X ap + bp = dk, a+b Cpi (dh)p−i (−1)i ai i=0 Cpi (dh)p−i (−1)i ai . (1-3) i=0 5¿ (a, b) = 1, d Ø a + b, ¤± (d, a) = 1. l d (1-3) ªáǑí Ñ d | p, ¤± d = 1 ½ö p. u´¤ Úny². y3·‚/Ïuù‡Ún5¤½n 1.2 y². ÏǑ a Ú b ǑØÓê, ¤±·‚Œ a = d · a , b = d · b , (a , b ) = 1, a + b = d · (a + b ). d Smarandache ¼ê S(n) 5Ÿ 1 p p p 1 p 1 1 1 p 1 S (ap + bp ) = S (dp · (ap1 + bp1 )) ≥ S (ap1 + bp1 ) . (1-4) u´Ǒy²½n 1.2, 5¿ (1-4) ª, ؔ˜„5·‚Œb½ (a, b) = 1, a · b > 1. éu?¿ƒê q | n, ·‚k S(n) ≥ q q | S (q ) é¤k ê α ¤á. y3, ·‚ky² a + b ،UǑ p ˜. eØ,, K k a + b = p . du p Ǒۃê,  α = 2 ž, du a + b > 2 > p , ¤ ± α ≥ 3, d ¡ÚnØJíÑ a + b = p u, 1 ≤ k ≤ α − 2, (u, p) = 1. 2d α p p p p α p p p k α p p p p k = a + b = a + (up − a) = = pk+1 uap−1 + p−2 X i=0 6 p p−1 X Cpi up−i pk(p−i) (−1)i ai i=0 Cpi up−i pk(p−i) (−1)i ai 2 1˜Ù 'u Smarandache ¼ê ½ö p−2 X α p − Cpi up−i pk(p−i) (−1)i ai = pk+1 uap−1 . i=0 þª†>U p Ø, ´m>ØUa, gñ ! ¤± a + b ،UǑ p +b ˜. u´3ƒê q 6= p q Ø a + b . =Ò´ a + b ≡ 0 (mod q) ½ö (a · b) ≡ −1 (mod q). l k k+2 p p p p p p p (a · b)2p ≡ 1 (mod q). (1-5)  m ´ (a · b)  q I, Kd (1-5) 9I5Ÿ (ë©z [8] 9 [9]) m | 2p. u´ m õko«ŒU: m = 1, 2, p, 2p. w, m 6= 1, 2, p. ÏǑe m = 1, K a ≡ b (mod q), † (a, b) = 1 q Ø a + b gñ. e m = 2, K a · b ≡ −1 (mod q) ½ö a + b ≡ 0 (mod q). †Ún 9 q Ø aa ++ bb gñ. 2d ¡Ó{ª (a · b) ≡ −1 (mod q)  m Ø ŒUu p, ¤±k m = 2p. 2dI5Ÿ m | φ(q) = q − 1, = Ò´ p p p p p q − 1 = h · m = h · 2p, ½ö q = h · 2p + 1. (1-6) u´d (1-6) ª a + b Ø p ƒ , –¹k 4 ‡ØÓƒÏfž, ˜½k˜‡ƒÏf q  p p q = h · 2p + 1 ≥ 4 · 2 · p + 1 = 8p + 1.  a + b ¹kn‡Øu p ƒÏf q , q 9 q ž, d (1-6) ª ·‚Œ q = 2h p + 1, q = 2h p + 1, q = 2h p + 1 h < h < h . dž h Ú h ،UӞǑ 1 Ú 2. eØ,, 5¿ p ≥ 11, K3 p, p = 2p + 1 Ú p = 4p + 1 n‡ƒê¥, –k˜‡U 3 Ø, ù† p, q Ú q ӞǑƒêgñ. Ïd h , h 9 h ¥–k˜‡Ø” h Œ u½u 4, dž·‚k q = 2h p + 1 ≥ 8p + 1. e¡·‚?Ø a + b ¹kü‡Øu p ƒÏf q œ¹. dž d¼ê S(n) 5ŸŒ·‚õIÄe¡ü«/ª: p p 1 1 1 1 1 1 2 2 3 2 3 3 1 2 3 2 2 2 1 p 3 p 2 3 3 3 7 'uSmarandache¯KïÄ#? a + b = p · (2p + 1) · (6p + 1) ½ö a + b = p · (4p + 1) · (6p + 1) . e a + b = p · (2p + 1) · (6p + 1) ¤á, K β ≥ 4 ½ö γ ≥ 2 ž, d S(n) 5ŸŒ: p p α p β p γ α β p p α β γ γ
S(ap + bp ) ≥ S (2p + 1)β = β · (2p + 1) ≥ 4 · (2p + 1) = 8p + 3 ≥ 8p + 1, ½ö S(ap + bp ) ≥ S ((6p + 1)γ ) = γ · (6p + 1) ≥ 2 · (6p + 1) = 12p + 2 ≥ 8p + 1. u´·‚Œb½ 1 ≤ β ≤ 3, γ = 1. y3·‚y²3ù«œ¹e  p ≥ 17 ž, a + b ،U¹k p ˜. eØ,,  α ≥ 2 ž, du p Ø a + b,  a + b = p · u, (p, u) = 1. Kd ¡Ún k = α ½ ö α − 1. w,d a + b = p · (2p + 1) · (6p + 1)  k = α ،U¤á, ÏǑdž p Ø a + b . u´Œ k = α − 1. l d a + b = p · u Œ p p k p p α+1 2·  pα−1 2 α p p ≤2·  β γ p α−1 a+b 2 p ≤ ap + bp = pα · (2p + 1)β · (6p + 1)γ . 5¿, α ≥ 2, 1 ≤ β ≤ 3, γ = 1, ¤± p ≥ 17 žN´yþªw ,ؤá.  α = 1 ž, du a + b ≡ a + b (mod p), ¤± k = α = 1, d ¡ ndŒ p | a + b , ù´ØŒU. ¤± a + b ،U¹kƒÏf p. ù·‚áǑ p 2 p p p p p 2p + 1 ≤ ap + bp = (2p + 1)β · (6p + 1), Ù¥ 1 ≤ β ≤ 3.  p ≥ 17 ž, ²LOŽþªØª´ØŒU¤á. ÓnŒ±y² p ≥ 17 ž, a + b = p · (4p + 1) · (6p + 1) β = γ = 1 ´ØŒU.  β ≥ 2 ½ö γ ≥ 2 ž, d S(n) 5Ÿ  S(a + b ) ≥ 8p + 1 ´w,. y3·‚?Ø a + b =¹k˜‡Øu p ƒÏf q œ¹. · ‚I?Ø: a + b = p · (2p + 1) , ½ö a + b = p · (4p + 1) , ½ö p p α p α p β p p α ap + bp = pα · (6p + 1)β . 8 β p p p p β γ ea p + bp = pα 1˜Ù 'u Smarandache ¼ê · (2p + 1) ¤á, K β ≥ 4 ž, w,kOª: β
S (ap + bp ) ≥ S (2p + 1)4 = 4 · (2p + 1) ≥ 8p + 1.  β ≤ 3 ž, d ¡y²L§Œ p ≥ 17 ž, e α ≥ 1, K a + b = p ·(2p+1) ،U¤á: Óe α = 0, K a +b = (2p+1) 1≤β≤3 Ǒؤá. ÓnŒy²1Ú1n«œ¹ a + b = p · (4p + 1) 9 a + b = p · (6p + 1) . u´¤ ½ny². α β p p α p p p p p p β α β β 1.3 Smarandahce 1.3.1 ¼ê3¤êêþe.O ¤êê†Ì‡(Ø é?¿šKê n, Ͷ¤êê F ½ÂǑ F = 2 + 1. ~ X F = 3, F = 5, F = 17, F = 257, F = 65537, · · · . w, 5 ‡¤ êêÑ´ƒê, u´¤êÒßÿé¤kšKê n, F Ǒƒê.  a¬êÆ[î.u 1732 ÞÑ ‡~: F = 641 × 6700417. Ïd¤ êߎ´†. ¯¢þ n = 6, 7, 8, 9, 11, 12, 18, 23, 36, 38, 73 ž, F ÑØ´ƒê. XJ F Ǒƒê, ·‚r§¡Ǒ¤êƒê. ´Ä3 ¡õ‡¤êƒê´˜‡™)ûêØJK. IêÆ[pdQy²: e F ´ƒê, K F >/Œ^ 59†ºŠÑ. ¤±¤êƒêǑk X­‡AÛµ. 'u Smarandache ¼ê3¤êêþe. O, <a3©z [7] ¥?1 ïÄ, ¼ ˜‡re.O. C, Á¯ [11] |^{!Š5Ÿ±9|ÜE|U? ©z [7] ¥ (Ø, ¼ e.O. äN/`ǑÒ´y² e¡: ½n 1.3. é?¿ê n ≥ 3, ·‚kOª n 0 1 2 3 2n n 4 n 5 n n n n S (Fn ) ≥ 12 · 2n + 1. 9 'uSmarandache¯KïÄ#? ½n 1.3 y² 1.3.2 ù!·‚^{!Š5Ÿ±9|ÜE|†‰Ñ½n 1.3 y². Äk5¿ F = 257, F = 65537, §‚Ñ´ƒê. Ïdé n = 3, 4, ·‚k S (F ) = 257 ≥ 12 · 2 + 1, S (F ) = 65537 > 12 · 2 + 1. Ïd ؔ˜„5·‚b½ n ≥ 5. XJ F = p, ˜‡ƒê, od S(n) 5 Ÿ·‚k S (F ) = S(p) = p = F = 2 + 1 ≥ 12 · 2 + 1; XJ F ´˜ ‡EÜê, o p ´ F ?¿ƒÏf, w, (2, p) = 1.  m L« 2 mod p I. =Ò´, m L«ê r  3 4 3 3 4 4 n n 2n n n n n 2r ≡ 1 (mod p). ÏǑ p | F , ·‚k F = 2 + 1 ≡ 0 (mod p) ½ö 2 ≡ −1 (mod p), 9 2 ≡ 1 (mod p). ddÓ{ª9I5Ÿ (ë©z [8] ¥½ n 10.1) ·‚k m | 2 , Ïd m ´ 2 ˜‡Ïf.  m = 2 , Ù ¥ 1 ≤ d ≤ n + 1. w, p ∤ 2 − 1, XJ d ≤ n. Ïd m = 2 ± 9 m | φ(p) = p − 1. u´ 2 | p − 1 ½ö n 2n n 2n 2n+1 n+1 n+1 d 2d n+1 n+1 p = h · 2n+1 + 1. (1-7) y3·‚©en«œ¹?Ø: (A). XJ F k–n‡ØÓƒÏf, Šâ (1-7) ªØ”Ǒ p = h ·2 + 1, i = 1, 2, 3. ÏǑ 2 +1 Ú 2·2 + 1 ،UӞǑƒ ê (–k˜‡U 3 Ø), 2 + 1 Ú 5 · 2 + 1 ،UӞǑƒ ê (–k˜‡U 3 Ø), 2 · 2 + 1 Ú 4 · 2 + 1 ،UӞǑ ƒê (–k˜‡U 3 Ø), 2 + 1 Ú 4 · 2 + 1 ،UӞǑƒ ê (–k˜‡U 3 ½ö 5 Ø), 2 · 2 + 1 Ú 3 · 2 + 1 ،UÓ žǑƒê (–k˜‡U 3 ½ö 5 Ø), 4 · 2 + 1 Ú 5 · 2 + 1 Ø ŒUӞǑƒê (–k˜‡U 3 Ø), ù˜5, 3 F ¤¹ 3 ‡ ØӃÏf¥, –k˜‡ p = h · 2 + 1 ¥ h ≥ 6. ؔ h ≥ 6, Kd S(n) 5Ÿ: n i i n+1 n+1 n+1 n+1 n+1 n+1 n+1 n+1 n+1 n+1 n+1 n+1 n+1 n i i n+1 i 3 S(Fn ) ≥ p3 ≥ 6 · 2n+1 + 1 = 12 · 2n + 1. XJ F T¹ü‡ØÓƒÏf, ؔ˜„5Œ


+1 · 5·2 2 +1 · 3·2 + 1 , ½ö 2 · 2 ½ö 3 · 2 + 1
· 4 · 2 + 1
. (B). Fn = n+1 n α β n+1 n+1 10 α n+1 α n+1 β n+1
β +1 , 1˜Ù 'u Smarandache ¼ê XJ F = 2 + 1
· 3 · 2 + 1
α ≥ 6 ½ö β ≥ 2, od S(n) 5Ÿ·‚áǑíÑOª α n+1 n n+1 β n 
α

β o S(Fn ) ≥ max S 2n+1 + 1 , S 3 · 2n+1 + 1 

= max α · 2n+1 + 1 , β · 3 · 2n+1 + 1 ≥ 12 · 2n + 1. XJ F = 2 + 1 = 2 + 1
· o5¿ n ≥ 5, ·‚kÓ{ª 2n n n+1
3 · 2n+1 + 1 = 3 · 22n+2 + 2n+3 + 1,  n 0 ≡ 22 + 1 − 1 = 3 · 22n+2 + 2n+3 ≡ 2n+3 (mod 2n+4 ).

n . , Fn = 22 + 1 6= 2n+1 + 1 · 3 · 2n+1 + 1 .
2
Fn = 2n+1 + 1 · 3 · 2n+1 + 1 = 3 · 23n+3 + 3 · 22n+3 + 3 · 2n+1 + 22n+2 + 2n+2 + 1, gñ Ïd XJ o·‚E,k n 0 ≡ 22 + 1 − 1 = 3 · 23n+3 + 3 · 22n+3 + 3 · 2n+1 + 22n+2 + 2n+2 ≡ 3 · 2n+1 (mod 2n+2 ).
2
n . , Fn = 22 + 1 6= 2n+1 + 1 · 3 · 2n+1 + 1 .
3
n Fn = 22 + 1 = 2n+1 + 1 · 3 · 2n+1 + 1 ,
2 n 22 + 1 ≡ 3 · 2n+1 + 1 ≡ 3 · 2n+2 + 1 (mod 2n+4 ), gñ Ïd XJ o ½ö
2 n 0 ≡ 22 ≡ 3 · 2n+1 + 1 − 1 ≡ 3 · 2n+2 (mod 2n+4 ). ù† 2 ∤ 3 · 2 gñ. XJ F = 2 + 1 = n+4 n+2 o
4
2n+1 + 1 · 3 · 2n+1 + 1 ,
4
≡ 2n+1 + 1 · 3 · 2n+1 + 1 − 1 ≡ 3 · 2n+1 (mod 2n+3 ). n n 0 ≡ 22 2n ù† 2 ∤ 3 · 2 gñ. XJ F = 2 + 1 = 2 n+3 n+1 n n 0 ≡ 22 n+1 ù† 2 ∤ 2 gñ, ÏǑ
n ≥ 5. XJ F = 2 · 2 + 1 · 5 · 2 od S(n) 5Ÿk 2n+2 o
5
+ 1 · 3 · 2n+1 + 1 ,
5
≡ 2n+1 + 1 · 3 · 2n+1 + 1 − 1 ≡ 2n+4 (mod 22n+2 ). 2n n+4 n n+1 n S(Fn ) ≥ max S α n+1
β +1 α≥3 ½ö β ≥ 2,  
α

β o 2 · 2n+1 + 1 , S 5 · 2n+1 + 1 11 'uSmarandache¯KïÄ#? 

= max α · 2 · 2n+1 + 1 , β · 5 · 2n+1 + 1 ≥ 12 · 2n + 1. XJ F o·‚k

n = 22 + 1 = 2 · 2n+1 + 1 · 5 · 2n+1 + 1 , n n Fn = 22 + 1 = 5 · 22n+3 + 7 · 2n+1 + 1. l ŒíÑÓ{ª n 0 ≡ 22 = 5 · 22n+3 + 7 · 2n+1 ≡ 7 · 2n+1 (mod 22n+3 ). ù´ØŒU, ÏǑ 2 XJ F = 2 + 1 = ª 2n+3 2n n ∤ 7 · 2n+1 .
2
2 · 2n+1 + 1 · 5 · 2n+1 + 1 , o·‚kÓ{
2
n 0 ≡ 22 = 2 · 2n+1 + 1 · 5 · 2n+1 + 1 − 1 ≡ 5 · 2n+1 (mod 2n+3 ). ù´ØŒU, ÏǑ 2 ∤ 5 · 2 . (C). XJ F Tk˜‡ƒÏf, ùž F Ǒƒêž, ½n 1.3 w ,¤á. u´·‚b½
F = 2 +1 ½ö F = 2 · 2 + 1
, α ≥ 2. XJ F = 2 + 1
, o α ≥ 6 ž½n 1.3 w,¤á. XJ α =
1, 2, 3, 4 ½ö 5, odÓ{ªØJíÑgñ. Ïd F 6= 2 +1 , 1 ≤ α ≤ 5. XJ F = 2 · 2 + 1
, o α ≥ 3 žd S(n) 5ŸŒ ½n 1.3 w,¤á.
XJ α = 1, o F Ǒƒê, ½n 1.3 Ǒ¤á.  F = 2 · 2 + 1 ž, dÓ{ª n+3 n+1 n n α n+1 n α α n+1 n n+1 n n n+1 n n α n 2 n+1 n 0 ≡ 22 = (2n+2 + 1)2 − 1 ≡ 2n+3 (mod 22n+2 ). áǑíÑgñ. ÏǑ n ≥ 5 ž, 2 ∤ 2 . (ܱþn«œ¹, ·‚áǑ¤ ½n 1.3 y². 2n+2 12 n+3 n+1 α 1˜Ù 'u Smarandache ¼ê 1.4 Smarandache 1.4.1 ¼ê3 £þe.O  £0 ½Â 1.2. éuê n, /X n!±1 ê¡Ǒ £ (shifted factorial). C, J. Sándor Ú F. Luca [12] Šâ C. L. Stewart [13] k' £ n! + 1 ƒÏꐡ(Jy² : lim sup n→∞ S(n! + 1) ≥ 5.5. n (1-8) S(n! + 1) = ∞. n (1-9) Ӟ, ©z [12] „Šâ M. Murthy Ú S. Wong [14] k' abc− ߎb 5(Jy² : XJ abc− ߎ¤á, K lim inf n→∞ ùp abc− ߎ´d J. Oesterlé [15] Ú D. W. Masser [16] JÑ Ͷߎ: pƒê a, b, c ·Ü a + b = c ž, éu?¿ê ǫ, È abc ØӃÏêÈ rad(abc) ÷v c < C(ǫ)(rad(abc)) , Ù ¥ C(ǫ) ´=† ǫ k'ŒkOŽ~ê. ù´˜‡î8™)û JK (넩z [17] ¯K B19). C, HýS$^{y² ± e ^‡›(J: ½n 1.4.  n > 10 ž, 1+ǫ 3   log n S(n! ± 1) , ≥ n log log n Ù¥ [α] L«¢ê α êÜ©. Šâþã½n†Œ±eíØ: ín 1.4.1. S(n! ± 1) lim n→∞ n = ∞. w,, íØ 1.4.1 Ø3ŸþU? (J (1-8), œ¹ey² (J (1-9). (1-10) (1-11) 3 ^‡› 13 'uSmarandache¯KïÄ#? , , $^þ¡½n„Œ±‰Ñk' £ƒÏêe..  P ´ n! + 1 ŒƒÏê. éd, P. Erdös Ú C. L. Stewart [18] y ² : 3 ¡õ‡ê n Œ P > 2n. d , ©z [13] ?˜Úy² : éu?Ûê ǫ, Œ P > (5.5 − ǫ)n ¤áê n äk—Ç. d , ©z [14] 3b½ abc− ߎ¤á^‡ey² : n n n lim inf n→∞ Pn = ∞. n Šâ!½nŒ±e(J: ín 1.4.2.  n > 10 ž, n! ± 1 7kƒÏê p ÷v 3   pr log n , ≥ n log log n (1-12) Ù¥ r ´ p 3 n! ± 1 IO©)ª¥gê. 1.4.2 ½nÚü‡íØy² Äk·‚0 A‡½ny²I‡Ún: Ún 1.4.1. XJ a = p · · · p ´ a IO©)ª, K S(a) = rk k r1 1 max{S(pr11 ), · · · , S(prkk }. Ún 1.4.2. éuƒê p Úê r, 7k p ≤ S(p ) ≤ pr. Ún 1.4.1 ÚÚn 1.4.2 y²ëì©z [19]. Ún 1.4.3.  x , x , · · · , x ´ k (k > 1) ‡™½. éu r ê m, 1 2 (x1 + x2 + · · · + xk )m = Ù¥ “P” L«é§ k X m n1 , n2 , · · · , nk ! xn1 1 xn2 2 · · · xnk k , n1 + n2 + · · · + nk = m, ni ∈ Z, ni ≥ 0, i = 1, 2, · · · , k 14 1˜Ù 'u Smarandache ¼ê ¤k) (n , n , · · · , n ) Ú, 1 2 k m n1 , n2 , · · · , nk ! = m! n1 !n2 ! · · · nk ! Ñ´ê, ¡Ǒõ‘ªXê. y²: 넩z [20] 1 1.2.2 !½n B. Ún 1.4.4.  x Ú y ´·Ü (x + 1)x+1 > ê.  y > 10 ž, 7k y e (1-13) 3 (x + 1) > Œ log y . log log y (1-14) y²: XJ x + 1 ≤ (log y)/ log log y, K3 (1-13) ü>éê log y (log log y − log log log y) > log y − 1. log log y (1-15) log log y > (log y)(log log log y). (1-16)  y > 10 ž, ÏǑ log log y > 0, ¤±l (1-15) Œ 3  z = log log y, l (1-16) Œ z z 1 log log y = z < = . 2 log y e z + z /2 1 + z/2 (1-17) Ïdl (1-16) Ú (1-17) Œ z 1 > (1 + ) log z. 2 (1-18) , , du y > 10 , ¤± z > 1.93 log z > 0.65, l (1-18) Œ  1 > 1.27 ù˜gñ. ddŒ:  y > 10 ž, ت (1-14) ¤á. Ú ny. ±e·‚é½nÚíØ?1y². 3 3 15 'uSmarandache¯KïÄ#? ½n 1.4 y²:  m = S(n! ± 1). Šâ Smarandache ¼ê½ Â: S(n) = min{k : k ∈ N, n | k!}, Œ m! = (n! ± 1)a, a ∈ N. (1-19) l (1-19) Œ m > n.  q = [m/n]. dž q 7Ǒê, Ø{Œ  |^‘{ m = nq + s, s ∈ Z, 0 ≤ s < n. (1-20) b = s!(n!)q . (1-21) ÏǑl (1-20) Ú (1-21) Œ ! m s, n, · · · , n m! m! = = b s!n! · · · n! (1-22) ´õ‘ªXê, ¤±lÚn 1.4.3 Œ m!/b ´ê. Ӟ, 3Ún 1.4.3 ¥ x = x = · · · = x ±9 k = q + 1, KŠâÚn 1.4.3, l (1-20) Ú (1-22) Œ 1 2 ,˜¡, ÏǑl 1) = 1. du®²y² k m! < (q + 1)m . b (1-20) s < n, b | m!, (1-19) Œ l (1-23) ¤±l (1-21) Œ gcd(b, n! ± Œ b | a. Ïdl (1-19) Œ m! a = (n! ± 1) ≥ n! ± 1. b b (1-24) (q + 1)m ≥ n!. (1-25) (Ü (1-23) Ú (1-24) á Šâ Stirling úªŒ n! > (n/e) , ¤±l (1-25) Œ n n (q + 1)m > ( )n . e (1-26) n . e (1-27) dul (1-20) Œ m < n(q + 1), ¤±l (1-26) Œ (q + 1)q+1 > 16  n > 10 3 1˜Ù 'u Smarandache ¼ê ž, ŠâÚn 1.4.4, l (1-27) Œ log n . log log n (1-28)  log n . q≥ log log n (1-29) n! ± 1 = pr11 · · · prkk (1-30) q+1> qÏ q ´ê, l (1-28) á  ÏǑl (1-20) Œ m ≥ nq, l (1-29) Œ (1-10). ½ny. íØ 1.4.1 y²:  ´ n! ± 1 IO©)ª, q p ´ (1-30) ¥Œ S(p ) Œƒê ˜, = S(p ) = max{S(p ), · · · , S(p )}. ŠâÚn 1.4.1 Œ r r r1 1 r rk k S(n! ± 1) = S(pr ). (1-31) S(n! ± 1) < pr. (1-32) qlÚn 1.4.2 Œ S(p ) < pr, l (1-31) Œ r u´, Šâ©½n¤(Ø (1-10), l (1-32) Œ (1-12) ¤á. íØ y. 17 'uSmarandache¯KïÄ#? 1Ù 'u Smarandache LCM ¼ê˜ ¯K Úó 2.1 Ùò0 ˜X'u Smarandache LCM ¼ê9§éó¼ê #ïĤJ, Äk·‚5wü‡½Â: ½Â 2.1. é?¿ê n, Ͷ Smarandache LCM ¼ê SL(n) ½ÂǑê k,  n | [1, 2, · · · , k], = SL(n) = min{k : k ∈ N, n | [1, 2, · · · , k]}, ùp [1, 2, · · · , k] L« 1, 2, · · · , k úê. ~X, SL(6) = 3, SL(10) = 5, SL(12) = 4, SL(20) = 5, · · · . AO n IO©)ª Ǒ n = p p · · · p ž, ØJy α1 α2 1 2 αk k SL(n) = max{pα1 1 , pα2 2 , · · · , pαk k }. ½Â 2.2. ·‚½Â¼ê SL(n) éó¼ê SL(n) Xeµ SL(n) = min{pα1 1 , pα2 2 , · · · , pαk k }. ~X, ù‡¼ê A‘Ǒ SL(1) = 1, SL(6) = 2, SL(12) = 3, SL(20) = 4, · · · . 2.2 'u Smarandache LCM ¼ê9Ùéó¼ê 'u SL(n) ù‡¼ê5Ÿ, NõÆö?1 ïÄ, ¿ Ø ­‡(J, ë©z [21-25]. ~X, èR [21] ïÄ SL(n) Š© Ù¯K, y² ìCúª: 2 (SL(n) − P (n))2 = · ζ 5 n≤x X 18   5 x2 5 · +O 2 ln x 5 x2 ln2 x ! , 1Ù 'u Smarandache LCM ¼ê˜ ¯K Ù¥ P (n) L« n ŒƒÏf. Le Maohua [22] ?Ø § SL(n) = S(n) Œ)5, ¿)û T¯K. =Ò´y² : ?Û÷vT§êŒL«Ǒ n = 12 ½ ö n = p p · · · p p, Ù¥ p , p , · · · , p , p ´ØÓƒê α , α , · · · , α ´÷v p > p , i = 1, 2, · · · , r ê. ë [23] ïÄ þŠ SL(n) − Ω(n)
ìC5Ÿ, ‰Ñ ì Cúªµ α1 α2 1 2 αr r αi i r 1 2 r 1 2 2 X n≤x
2 4 SL(n) − Ω(n) = ζ 5   5 5 k X x2 5 ci · x 2 +O · + 4 ln x i=2 lni x 5 x2 lnk+1 x ! , Ù¥ ζ(n) Ǒ Riemann zeta- ¼ê, c ǑŒOŽ~ê, Ω(n) ǑŒ\¼ê, ½ÂǑ: Ω(n) = X α p , XJ n IO©)ªǑ n = p p · · · p . i k α1 α2 1 2 j j αk k j=1 'u SL(n) ù˜#¼ê5Ÿ, ·‚–8$, $– اþŠ©Ù5Ÿ. !̇8´0 A¡_ [26] óŠ, = |^9|ܐ{ïÄ·ÜþŠ X SL(n) n≤x (2-1) SL(n) ìC5Ÿ, ¿y² e¡(Ø: ½n 2.1. é?¿¢ê x > 1, ·‚kìCúªµ X SL(n) n≤x x = +O SL(n) ln x  x(ln ln x)2 ln2 x  . w,ù‡½n¥Ø ‘´š~f, ǑÒ´`Ø ‘†Ì‘= ˜‡ (lnlnlnxx) Ïf, ´Ä3 (2-1) ª˜‡rìCúªǑ´˜‡ k¯K, ïÆk,Öö?˜ÚïÄ. y²: ±e·‚†‰Ñ½n 2.1 y². ¯¢þ·‚ò¤ku½ u x ê n ©Ǒ±en«œ¹?Ø: A = {n : ω(n) = 1, n ≤ x}; B = {n : ω(n) = 2, n ≤ x}; C = {n : ω(n) ≥ 3, n ≤ x}, Ù¥ ω(n) L 2 19 'uSmarandache¯KïÄ#? « n ¤kØӃÏf‡ê. y3·‚©OO¼ê SL(n) 3ùn SL(n) ‡8ÜþþŠ. 5¿ƒê½n [27] π(x) = X p≤x ·‚k X SL(n) = SL(n) n∈A = x +O 1= ln x   x , ln2 x X SL(pα ) X SL(p) X SL(pα ) = + SL(pα ) SL(p) SL(pα ) α α p ≤x X p≤x 1+ p≤x X 1= pα ≤x α≥2 x +O = ln x  x ln2 x p ≤x α≥2 x +O ln x   x ln2 x    X +O  X  1 1 2≤α≤ln x p≤x α . (2-2) y3·‚OÌ‡Ø ‘.  n ∈ B ž, ·‚k n = p q , Ù¥ p 9 q ǑØÓƒê. ؔ p < q , 5¿ìCúª (ë©z [8]) α β α X1 p≤x ·‚k X SL(n) SL(n) = n∈B p β = ln ln x + C1 + O X p + qβ √ α X pα qβ p ≤ x pα <q β ≤ pxα X X p ≤ x pα <q β ≤ pxα α≥2       X X X p X 1   p 2 ln x + O  pα ln ln x +O  q  √ √ √ x α X √ √ x <p≤ x ln x 20 , pα qβ p≤ x p<q≤ p =  X √ p≤ x p<q β ≤ xp = 1 ln x X SL(pα ) X = SL(q β ) √ α β α p q ≤x pα <q β =  p≤ x p ≤ x α≥2   X Xp X p x + +O ln2 x √ x q x q x p<q≤ p p≤ ln x q≤ p 1Ù 'u Smarandache LCM ¼ê˜ ¯K = X      x ln ln x 1 x +O p ln ln − ln ln p + O p ln x ln2 x X ln x − ln p p ln +O ln p √ √ x <p≤ x ln x = √ √ x <p≤ x ln x  x ln ln x ln2 x  ln x − 21 ln x + ln ln x x ln ln x ≪ + p ln 1 ln2 x √ √ 2 ln x − ln ln x x <p≤ x ln x   X x ln ln x 4 ln ln x + ≪ p ln 1 + ln x − 2 ln ln x ln2 x √ √ x X ln x <p≤ x X ≪ √ √ x <p≤ x ln x x ln ln x p ln ln x x ln ln x + ≪ . 2 ln x − 2 ln ln x ln x ln2 x (2-3)  n ∈ C ž, ·‚± ω(n) = 3 Ǒ~‰ƒy². ؔ n = p p p p < p < p . u´dþªO{·‚k β 2 α 1 X β γ pα 1 p2 p3 ≤x α β γ 1 2 3 γ 3 SL(pα1 ) SL(pγ3 ) X = 1 3 pα 1 ≤x β γ pα 1 <p2 <p3 X pα1 pβ2 pγ3 ≤ pxα 1 pβ2 <pγ3   X p1 = O 1 p1 ≤x 3 ≪ 1 pγ3 X √ X x p1 <p2 ≤ √px p2 <p3 ≤ p1 p2 1 X √xp1 x ln ln x ln ln x ≪ . ln x ln2 x 1  1  p3 (2-4) p1 ≤x 3 5¿ê n ¤kØӃÏf‡ê ω(n) ≪ ln ln n, u´‡EA ^ (2-4) ª·‚ØJíÑOª X SL(n) SL(n) = X n≤x 3≤ω(n)≤ln ln x n∈C ≪ x(ln ln x)2 . ln2 x SL(n) = SL(n) X 3≤k≤ln ln x X SL(n) SL(n) n≤x ω(n)=k (2-5) 21 'uSmarandache¯KïÄ#? y3(Ü (2-2)!(2-3) 9 (2-5) ª·‚íÑOª X SL(n) x = +O SL(n) ln x n≤x u´¤ ½n 2.1 y². 2.3 Smarandache LCM ÏfþŠ  x(ln ln x)2 ln2 x  . ¼êéó¼ê†ƒ þ!·‚?Ø Smarandache LCM ¼ê†Ùéó¼ê'Ç'X, ¼ ˜‡fþŠúª. !UY?Ø Smarandache LCM ¼ê éó¼êÙ§5Ÿ, ǑÒ´|^9)ې{ïÄþŠ X n≤x
2 SL(n) − p(n) (2-6) ìC5Ÿ§Ù¥ p(n) L« n ƒÏf. ~X p(20) = 2, p(21) = 3. 'u (2-6) ªþŠ5Ÿ, –8qvk<ïÄ, –·‚vk3yk ©z¥w. , , ù˜¯K´k¿Â, ÏǑ (2-6) ªìC5‡N ùü‡¼êŠ©Ù5Æ5. !éù˜¯K?1 ïÄ, ¿‰Ñ ˜‡kþŠúª. äN/`ǑÒ´y² e¡(Ø: ½n 2.2.  k Ǒ‰½ê. oé?¿¢ê x > 1, ·‚k ìCúª: [28] X n≤x k
2 X ci · x 2 SL(n) − p(n) = +O i ln x i=1 5 5 x2 lnk+1 x ! , Ù¥ c (i = 1, 2, · · · , k) ´ŒOŽ~ê c = 54 . w,½n 2.2 ¥Ø ‘´š~f, ǑÒ´`Ø ‘†Ì‘= 1 ˜‡ ln x Ïf, ´Ä3 (2-6) ª˜‡rìCúªǑ´˜‡k ¯K. ïÆk,Öö?˜ÚïÄ. i 22 1 1Ù 'u Smarandache LCM ¼ê˜ ¯K y²: e¡·‚(Ü9)ې{†‰Ñ½n 2.2 y². ¯¢þ·‚ò¤ku½u x ê n ©Ǒ±eü‡8Ü?Ø: A = {n : ω(n) = 1, n ≤ x}; B = {n : ω(n) ≥ 2, n ≤ x}, Ù¥ ω(n) L « n ¤kØӃÏf‡ê. y3·‚©OO¼ê SL(n) − p(n)
3ùü‡8ÜþþŠ. 5¿é?¿ê k, d©z [27] ¥½n 3.2 Œ   2 π(x) = X 1= p≤x Ù¥ a ǑŒOŽ~ê z [8] ¥½n 4.2) Œ p4 = √ p≤ x = = i ln x i=1 a1 = 1. i X k X ai · x √
x2 · π x − 3 +O Z √ x x , 2 y 3 · π(y)dy  √ ! √ k X x x a · i +O x2 i√ k+1 ln x ln x i=1 !  Z √x k X a · y y i dy −3 +O y3 · i k+1 ln y ln y 2 i=1 ! 5 5 k X ci · x 2 x2 +O , (2-7) lni x lnk+1 x i=1 4 c1 = . 5 i n∈A ln u´A^ Abel Úúª (ë© Ù¥ c (i = 1, 2, · · · , k) ǑŒOŽ~ê u´A^ (2-7) ª·‚k X x k+1
2 SL(n) − p(n) = = X pα ≤x X p≤x =
2 SL(pα ) − p (p − p)2 + X X pα ≤x α≥2 (p2 − p)2 + √ p≤ x = X √ p≤ x  p4 + O  (pα − p)2 X pα ≤x α≥3 X √ p≤ x (pα − p)2     X 6 p3  + O  p  1 p≤x 3 23 'uSmarandache¯KïÄ#? = 5 k X ci · x 2 5 +O lni x i=1 x2 lnk+1 x ! . (2-8) y3·‚OÌ‡Ø ‘.  n ∈ B ž, du ω(n) ≥ 2, ·‚Ø”  SL(n) = q , n = q n , Ù¥ SL(n ) > q . XJ α = 1, o SL(n) −
p(n) = 0. u´ SL(n) − p(n) 6= 0 žk α ≥ 2 kت q < √ n < n , l A^ Abel Ú·‚ØJ: α α 1 α 1 2 α 1 X n∈B
2 SL(n) − p(n) = ≪ X √ qα ≤ x X √ (q α − p(qn1 ))2 n1 ≤ qxα SL(n1 )>q α X n1 ≤ xq q≤ x ≪ x X (q − q)2 + SL(n1 )>q X 1 q≤x 4 X X q6 x 1 q≤x 4 n1 ≤ q 2 5 4 9 4 q ≪x ≪ x2 lnk+1 x . (2-9) y3(Ü (2-8) 9 (2-9) ª·‚áǑíÑìCúª X n≤x X
2
2 X
2 SL(n) − p(n) = SL(n) − p(n) + SL(n) − p(n) n∈A = n∈B k X ci · x i=1 5 2 lni x +O Ù¥ c (i = 1, 2, · · · , k) ǑŒOŽ~ê ². i 2.4 x 5 2 lnk+1 x ! 4 c1 = . 5 , u´¤(Øy ˜‡¹ Smarandache LCM ¼êéó¼ ꐧ ü!·‚̇ïÄ † Smarandache LCM ¼êéó¼ ê SL(n) k'þŠ5Ÿ, ù˜!·‚?ؘ‡¹ SL(n) §, ?˜Ú&¢ Smarandache LCM ¼êéó¼êŽê5Ÿ. 24 1Ù 'u Smarandache LCM ¼ê˜ ¯K ïÄuy¼ê SL(n) †¼ê SL(n) kNõƒq5Ÿ, ~X,  n Ǒƒê˜ž, SL(n) = SL(n). éu SL(n) ¼ê9î.¼ê ϕ(n), ² P u·‚uy3 õ‡ê n  SL(d) > ϕ(n). ¯¢þ, ´ íÑ n = p Ǒƒê˜ž, ·‚k d|n α X SL(d) = X d|pα d|n SL(d) = 1 + p + · · · + pα > pα − pα−1 = ϕ(n), Ӟq3 õ‡ê n,  P SL(d) < ϕ(n). ~X,  n Ǒü ‡ØÓۃêȞ, = n = p · q, e 5 ≤ p < q Ǒƒê, o d|n X SL(d) = X d|p·q d|n SL(d) = 1 + 2p + q < (p − 1) · (q − 1) = ϕ(n). u´·‚g,Ž, éu= ê n ¬k§ X SL(d) = ϕ(n). (2-10) d|n ¤á, Ù¥ X L«é n ¤kÏêÚ, ϕ(n) Ǒî.¼ê. d|n !̇8Ò´0 I [29] ¤J, =|^{ïÄ § (2-10) Œ)5, ¿¼ T§¤kê). äN/`Ò´ y² e¡: ½n 2.3. § X SL(d) = ϕ(n) k =kʇê) n = d|n 1, 75, 88, 102, 132. Ǒ ¤½ny², ÄkI‡ü‡{üÚn. Ún 2.4.1. ت ϕ(n) < 4d(m) ¤á = m = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 32, 36, 40, 42, 48, 56, 60, 72, 80, 84, 96, 120, 144, 168, 288. ùp d(m) Ǒ Dirichlet Øê¼ê. y²: - m = p p · · · p L« m IO©)ª. ·‚©±eA «œ¹5?1?Ø: α1 α2 1 2 αk k 25 'uSmarandache¯KïÄ#? i) XJ©)ª¥3Ïf 2 α ≥ 6, Kk α αi 1 k k ϕ(m) Y pi (1 − pi ) Y piαi −1 (pi − 1) 2α−1 = = ≥ > 4, d(m) α + 1 α + 1 α + 1 i i i=1 i=1 = ϕ(m) ≥ 4d(m). ii) XJ©)ª¥3Ïf 3 α α ≥ 3, Kk ϕ(m) 3α−1 · 2 ≥ > 4, d(m) α+1 = ϕ(m) ≥ 4d(m). iii) XJ©)ª¥3Ïf 5 α α ≥ 2, Kk ϕ(m) 5α−1 · 4 ≥ > 4, d(m) α+1 = ϕ(m) ≥ 4d(m). iv) XJ©)ª¥3Ïf 7 α α ≥ 2, Kk 7α−1 · 6 ϕ(m) ≥ > 4, d(m) α+1 = ϕ(m) ≥ 4d(m). v) XJ©)ª¥3Ïf p α p ≥ 11, Kk ϕ(m) pα−1 · (p − 1) ≥ > 4, d(m) α+1 = ϕ(m) ≥ 4d(m). Ïd·‚I3 m = 2 · 3 · 5 · 7 (0 ≤ α ≤ 5, 0 ≤ β ≤ 2, γ = δ = 0 ½ 1) ¥Ïé÷v^‡ ϕ(m) < 4d(m)  ê m =Œ, ²Ly, ѱe 35 ‡÷v^‡ m : m = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 32, 36, 40, 42, 48, 56, 60,72, 80, 84, 96, 120, 144, 168, 288. u´¤ Ún 2.4.1 y². Ún 2.4.2.  m عkƒÏf 2 ž, ت ϕ(m) < 6d(m) ¤á  = m = 1, 3, 5, 7, 9, 11, 15, 21, 27, 33, 35, 45, 63, 105. y²: - m = p p · · · p L« m IO©)Ϫ, Ù¥ p ≥ 3 (i = 1, 2, · · · , k). ·‚©±eA«œ¹5?1?Ø: α α1 α2 1 2 26 αk k β γ δ i 1Ù 'u Smarandache LCM ¼ê˜ ¯K i) XJ©)ª¥3Ïf 3 α ≥ 4, Kk α ϕ(m) 3α−1 · 2 ≥ > 6, d(m) α+1 = ϕ(m) ≥ 6d(m). ii) XJ©)ª¥3Ïf 5 α α ≥ 2, Kk ϕ(m) 5α−1 · 4 ≥ > 6, d(m) α+1 = ϕ(m) ≥ 6d(m). iii) XJ©)ª¥3Ïf 7 α α ≥ 2, Kk ϕ(m) 7α−1 · 6 ≥ > 6, d(m) α+1 = ϕ(m) ≥ 6d(m). iv) XJ©)ª¥3Ïf 11 α α ≥ 2, Kk ϕ(m) 11α−1 · (p − 1) ≥ > 6, d(m) α+1 = ϕ(m) ≥ 6d(m). v) XJ©)ª¥3Ïf p α p ≥ 13, Kk pα−1 · (p − 1) ϕ(m) ≥ ≥ 6, d(m) α+1 = ϕ(m) ≥ 6d(m). Ïd·‚I3 m = 2 · 3 · 5 · 7 (0 ≤ α ≤ 5, 0 ≤ β ≤ 2, γ = δ = 0 ½ 1) ¥Ïé÷v^‡ ϕ(m) < 6d(m)  ê m =Œ, ²Ly, ѱe 14 ‡÷v^‡ m : m = 1, 3, 5, 7, 9, 11, 15, 21, 27, 33, 35, 45, 63, 105. u´¤ Ún 2.4.2 y². ½ny²: y3·‚|^ùü‡Ún5‰Ñ½ny². N´ y n = 1 ´§).  n > 1 n = p p · · · p ´ n IO©) ª, ÏǑ n = p Ø÷v§, ¤± n ÷v§žk k ≥ 2. y3 α β γ α1 α2 1 2 δ αk k α SL(n) = max{pα1 1 , pα2 2 , · · · , pαk k } = pα . 27 'uSmarandache¯KïÄ#? ǑBå„ n = mp ÷v§, džAk: α X SL(d) = α X X i SL(dp ) = i=0 d|m d|n X SL(d) + SL(dpi ) i=1 d|m d|m = pα−1 (p − 1)ϕ(m). α X X ÏǑ d | m ž, SL(dp ) ≤ p , ¤± i pα−1 (p − 1)ϕ(m) ≤ = X i SL(d) + i=1 d|m α d|m X α X X SL(d) + d|m þªü>Óر p α−1 ϕ(m) ≤ ≤ X d|m (p − 1), pi = X d|m SL(d) + d(m) · p(p − 1) d(m). p−1 α X pi i=1 ¿5¿ d | m ž SL(d) ≤ p , ¤±k i p(pα − 1) SL(d) + d(m) pα−1 (p − 1) pα−1 (p − 1)2 p p 2 p(p − 1) + p2 · d(m) + ( ) · d(m) = · d(m). p−1 p−1 (p − 1)2  p > 2 ž, þªCǑ ϕ(m) < 4ϕ(m),  p = 2 ž, þªCǑ ϕ(m) ≤ 6d(m). =e n = mp ÷v§,  p > 2 ž, Ak ϕ(m) < 4d(m), Ǒ Ò´ ϕ(m) ≥ 4d(m) ž, n = mp Ø´§); ½ p = 2 ž, A k ϕ(m) ≤ 6d(m), ǑÒ´ ϕ(m) > 6d(m) ž, n = m · 2 Ø´§ ). dÚn 2.4.1 Œ, ϕ(m) < 4d(m)  = m = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 32, 36, 40, 42, 48, 56,60, 72, 80, 84, 96, 120, 144, 168, 288. dÚn 2.4.2 Œ, p = 2 ž, ϕ(m) ≤ 6d(m)  = m = 1, 3, 5, 7, 9, 11, 15, 21, 27, 33, 35, 45, 63, 105. e¡I?Ø 3þãÞ m ¥,  n = mp ÷v§=Œ. 1)  m = 1 ž, n = p , ùp p ´?¿ƒê. α α α α α X d|pα SL(d) = 1 + p + p2 + · · · + pα > pα − pα−1 = ϕ(pα ), = n = p Ø´§ (2-10) ). α 28 1Ù 'u Smarandache LCM ¼ê˜ ¯K 2)  m = 2 ž, n = 2p , ùp p ≥ 3. α X SL(d) = d|2pα SL(d) + d|pα d|2pα X X X SL(2d) = d|pα X SL(d) + 2(α + 1) d|pα = 1 + p + p2 + · · · + pα + 2(α + 1), SL(d) > ϕ(2pα ) = ϕ(2)ϕ(pα ) = pα − pα−1 . ¤± n = 2p (p ≥ 3) Ø´§ (2-10) ). 3)  m = 3 ž, n = 3p , ùp p 6= 3, e p = 2, α α X SL(d) = X SL(d) + d|2α d|3·2α é α ^êÆ8B{Œy 2 X d|3·2α X SL(3d) = 2α+1 + 3α + 1. d|2α α+1 + 3α + 1 > 3 · 2α−1 , = SL(d) > ϕ(3 · 2α ). e p = 5,  α = 1 ž, n = 15 Ø´§ (2-10) );  α = 2 ž, n = 75 ÷v§ (2-10), Ï ´§ (2-10) );  α ≥ 3 ^êÆ8B {Œy X d|3·5α SL(d) = 1 + 5 + 52 + · · · + 5α + 3(α + 1) < 2(5α − 5α−1 ) = ϕ(3 · 5α ). dž n = 3 · 5 Ø´§ (2-10) ). e p ≥ 5 ž, ÓþŒy n = 3 · p Ø´§ (2-10) ). 4)  m = 4 ž, n = 4p , K p ≥ 3, © p = 3, p = 5, p = 7, p = 11, p = 13 9 p > 13 8«œ¹^þ¡{?Ø n = 4p ÑØ´ § (2-10) ). 5)  m = 5 ž, © p = 2, p = 3, p > 5 ?Ø n = 5p ÑØ´ § (2-10) ). 6)  m = 6 ž, n = 6p , K p ≥ 5. ²Ly p = 17, α = 1 ž, n = 102 ´§ (2-10) ), Ùœ¹ÑØ´§ (2-10) ). 7)  m = 7, 8, 9, 10 ž, ÓþŒy n = m · p ÑØ´§ (2-10) ). 8)  m = 11 ž, dÚn 2.4.2 , p = 2, dž n = 11 · 2 , N´ y α = 3 ž n = 88 ´§ (2-10) ), é α ÙŠ n ÑØ´ § (2-10) ). α α α α α α α α 29 'uSmarandache¯KïÄ#? 9)  m = 12 ž, K n = 12·p , dž p ≥ 5. N´y p = 11, α = 1 ž, n = 132 ´§ (2-10) ), é p 9 α ÙŠ n ÑØ´ § (2-10) ). 10)  m = 27, 33, 35, 45, 63, 105 ž, p = 2, Œ±yùž n = m · 2 ÑØ´§ (2-10) ). 11)  m = 14, 15, 16, 18, 20, 21, 24, 28, 30, 32, 36, 40, 42, 48, 56, 60, 72, 80, 84, 96, 120, 144, 168, 288, ÓþŒ±yùž n = m · p ÑØ´ § (2-10) ). nþ¤ã, § P SL(d) = ϕ(n) k =kʇê) n = 1, 75, 88, 102, 132. ùÒ¤ ½ny². α α α d|n 2.5 Smarandache ê·ÜþŠ ¼ê† Smarandache LCM ¼ 'u Smarandache ¼ê9 Smarandache LCM ¼ê˜ 5Ÿ, 3 ¡®²‰ ØïÄ. , , k'ùü‡¼êŠ'ǯKl™k<J . ǑÒ´þŠ X S(n) SL(n) (2-11) n≤x ìC5Ÿ. ù˜¯K´k¿Â, ÏǑ (2-11) ªìC5‡N ùü ‡¼êŠ©Ù5Æ5, XJìCúª X S(n) ∼x SL(n) n≤x ¤á, o·‚ÒŒ±ä½¼ê S(n) † SL(n) ŠA??ƒ. ! ò0 ²^ [30] (J, ǑÒ´éù˜¯K?1 ïÄ, ¿y² §(5. äN/`ǑÒ´y² e¡ü‡(Ø: ½n 2.4. é?¿¢ê x > 1, ·‚kìCúª:   X S(n) x ln ln x . =x+O SL(n) ln x n≤x 30 1Ù 'u Smarandache LCM ¼ê˜ ¯K ½n 2.5. é?¿¢ê x > 1, ·‚kìCúª:   X P (n) x ln ln x , =x+O SL(n) ln x n≤x Ù¥ P (n) L« n ŒƒÏf. ,ùü‡½n¥Ø ‘Ǒ´š~f, ´Ä3˜‡rì CúªǑ´˜‡k¯K. y²: ±e·‚ò†‰Ñ½ny². ·‚y²½n 2.4, aq , ·‚ǑŒ±íѽn 2.5. ¯¢þ²L{üC/·‚áǑ: X S(n) X S(n) − SL(n) X = + 1 SL(n) SL(n) n≤x n≤x = x+O n≤x X |SL(n) − S(n)| SL(n) n≤x ! . (2-12) y3·‚|^¼ê S(n) 9 SL(n) 5Ÿ±9†|ܐ{5O (212) ª¥Ø ‘. d SL(n) 5Ÿ n IO©)ªǑ p p · · · p žk: α1 α2 1 2 αk k SL(n) = max{pα1 1 , pα2 2 , · · · , pαk k }. e SL(n) Ǒƒê p, o S(n) ǑǑƒê p. Ïd, 3ù«œ¹e k SL(n) − S(n) = 0. ¤±3 (2-12) ªØ ‘¥, ¤kš"‘7 Ñy3  SL(n) Øuƒêê n ¥. ǑÒ´`µ SL(n) = max{pα1 1 , pα2 2 , · · · , pαk k } ≡ pα , α ≥ 2.  A Ǒ«m [1, x] ¥¤k÷vþª^‡ n 8Ü, é?¿ n ∈ A,  n = p p · · · p = p · n , Ù¥ (p, n ) = 1. y3·‚©ü«œ¹? Ø:  A = B + C, Ù¥ n ∈ B XJ SL(n) = p ≥ 9(lnlnlnxx) ; n ∈ C X J SL(n) = p < 9(lnlnlnxx) . u´·‚k α1 α2 1 2 αk k α 1 1 2 α 2 2 α 2 X |SL(n) − S(n)| n≤x SL(n) = X |SL(n) − S(n)| X |SL(n) − S(n)| + SL(n) SL(n) n∈B n∈C 31 'uSmarandache¯KïÄ#? ≤ X X 1+ X 1 n∈C x 9x(ln ln x)2 ln2 x ≤pα ≤ n n≤ ln2 x 9(ln ln x)2 α≥2 ≡ R1 + R2 . y3·‚©OO (2-13) ª¥ˆ‘. ÄkO R . 5¿ p žk α ≤ 4 ln ln x. u´dƒê½n·‚k α 1 X X R1 ≤ x n≤ lnx4 x pα ≤ n α≥2 X ≪ n≤ lnx4 x X ≪ n≤ lnx4 x p≤ X √ x α≤ln x n 1 X 1+ X ln x)2 x ≤n≤ 9x(ln ln4 x ln2 x X X ln x)2 x ≤n≤ 9x(ln ln4 x ln2 x x ln x + n ln ln x ≤ ln4 x x pα ≤ n α≥2 ln x)2 x ≤n≤ 9x(ln ln4 x ln2 x X r X X 1+ (2-13) p≤ r √ x α≤4 ln ln x n 1 x x ln ln x ≪ . (2-14) n ln x y3·‚O R , 5¿8Ü C ¥¹ƒ‡êج‡L ê p p · · · p ‡ê, Ù¥ α ≤ 2 ln ln x, p ≤ 3 lnlnlnx x , i = 1, 2, · · · . u´5¿ƒê©Ùúª     2 αk k α1 α2 1 2 ·‚k X i y ln y ln p = y + O p≤y R2 = X n∈C ≪ 1≤ Y ln x p≤ 3 ln ln x  Y ln x p≤ 3 ln ln x  i   − ln 1 − X −1 X ln x p≤ 3 ln ln x   α p 0≤α≤2 ln ln x 1 1− ln p 3 ≪ exp  ln x + 4 , 1 ln p ≤ exp 2 ln ln x  ∼ 1 , ln p Y ln x p≤ 3 ln ln x X ln x p≤ 3 ln ln x 1  x ≪ , ln p ln x p2 ln ln x 1 − ln1p  ln p (2-15) Ù¥ exp(y) = e . (Ü (2-13)!(2-14) 9 (2-15) ª·‚íÑOª y X |SL(n) − S(n)| n≤x 32 SL(n) ≪ x ln ln x . ln x (2-16) 1Ù 'u Smarandache LCM ¼ê˜ ¯K |^ (2-12) 9 (2-16) ªáǑíÑìCúª:   X S(n) x ln ln x . =x+O SL(n) ln x n≤x u´¤ ½n 2.4 y². 5¿ SL(n) = p Ǒƒêž, S(n) = P (n) = p;  SL(n) ØǑ ƒêž, P (n) ≤ S(n) ≤ SL(n), u´dy²½n 2.4 {áǑíѽ n 2.5. 2.6 ˜‡¹ Smarandache ¼ê† Smarandache LCM ¼ê§ LCM þ˜!·‚0 Smarandache ¼ê S(n) 9 Smarandache X S(n) ¼ê SL(n) ùü‡¼êŠ'ǯK, ǑÒ´þŠ SL(n) ìC 5Ÿ, l(JŒ±ä½¼ê S(n) † SL(n) ŠA??ƒ. S(n) † SL(n) Ø kù‡5Ÿƒ , „¬kŸoéXQ? !ò0  I [31] ïĤJ, äN`Ò´|^9|ܐ{ïЧ n≤x X d|n S(d) = X SL(d) (2-17) d|n Œ)5, ¿¼§¤kê). =y² e¡½n: ½n 2.6. ª (2-17) k ¡õ‡ ê), = n = 1, 2 p p · · · p , Ù¥ k ´˜‡?¿ê, α = 0, 1 ½ö 2, p , p , · · · , p ´ØÓƒê ÷v 2 < p < p < · · · < p .  A L«÷v§ (2-17) ¤kê n 8Ü, oŒ± e¡½n. ½n 2.7. éu?¿Eê s Re(s) > 1, k αk k α α1 α2 1 2 1 2 k 1 2 k X 1 ζ(s) 4s + 2s + 1 = , ns ζ(2s) 4s + 2s n∈A 33 'uSmarandache¯KïÄ#? Ù¥ ζ(s) ´ Riemann zeta- ¼ê. ½n 2.8. éu?¿¢ê x ≥ 1, kìCúª X 1= n≤x n∈A √ 7 x + O( x). π2 5¿ Ú u´d½n 2.7 Œ±Ñ e¡íØ íØ 2.6.1. 3½n 2.7 ¥, - s = 2, 4, Kkðª X 1 X 1 63 28665 = = . 9 n 4π n 272π π4 π2 ζ(2) = , ζ(4) = 6 90 . 2 π8 , ζ(8) = 9450 2 4 n∈A 4 n∈A ½ny²: ±e·‚5¤½ny². Äk, y²½n 2.6. ¯ ¢þ, d¼ê S(n) Ú SL(n) ½ÂŒ, n = 1 ´§ (2-17) ˜‡ ).  n IO©)ªǑ n = p p · · · p , d S(n) Ú SL(n) ½ÂÚ 5ŸŒ± S(n) = max{S(p ), S(p ), · · · , S(p )} = S(p ) ≤ α p , SL(n) = max{p , p , · · · , p } = p , w, p ≥ p ≥ α p . ¤±, é ?¿ê n > 1, - n = 2 p p · · · p (2 < p < · · · < p ), Œ±ò n ©¤n«œ¹?1?Ø: 1) é α = 0, 1, k (a) XJ α = α = · · · = α = 1, = n = p p · · · p ½ö n = 2p p · · · p , éu n ?¿Ïf d, k S(d) = SL(d), §‚´§ (217) ). (b) XJ–k˜‡ α (i ≥ 2), k S(p ) ≤ α p , SL(p ) = p ,  §‚Ø÷v§ (2-17). 2) é α = 2, α = α = · · · = α = 1, k (c) XJ p = 3, = n = 4 × 3n (12 ∤ n ), Kk α1 α2 1 2 α1 1 α1 1 1 1 2 αj αk j k α α1 α2 1 2 α2 2 2 α2 2 1 i i i i k k 1 2 αi i 2 S(d) = 1 X S(d) + d|n1 + X d|n1 X S(2d) + 1 X d|n1 d|n1 S(6d) + i i k 1 34 αi i αi i k k 1 d|n αk k αj j αk k i X αk k X d|n1 S(12d) S(4d) + X d|n1 S(3d) αi i αi i 1Ù 'u Smarandache LCM ¼ê˜ ¯K X = d|n1 S(d) + (2 − 1 + (3 − 1 + = 11 + 6 X d|n1 X X d|n1 S(d)) + (4 − 1 + S(d)) + (3 − 1 + X d|n1 X S(d)) + d|n1 S(d)) + (4 − 1 + X S(d)) d|n1 S(d), d|n1 Ӟk SL(d) = 11 + 6 X SL(d). Ä X S(d) = X SL(d), § ‚´§ (2-17) ). X (d) XJ p > 3, = n = 4 × n (4 ∤ n ), Œ± S(d) = X X X S(d) Ú SL(d), §‚´§ (2-17)  SL(d) = 4 + 3 4+3 ). 3) XJ α ≥ 3, Œ±–3˜‡gê α ≥ 2,  S(2 ) ≤ 2α, SL(2 ) = 2 > 2α, u´§‚Ø´§ (2-17) ). (ܱþA«?Øœ¹, Œ±§ (2-17) 3 êõ‡ ê), §‚´ n = 1, 2 p p · · · p ( α = 0, 1 ½ 2 ), Ù¥ 2 < p < · · · < p L«ØÓƒê, –d·‚¤ ½n 2.6 y². y3·‚5y²½n 2.7. lî.Èúª („©z [8] ½n 11.7) Ú$'¿d¼ê5Ÿ±9 Riemann zeta- ¼ê½Â, ·‚Œ± X d|n d|n1 d|n1 d|n1 1 1 1 d|n d|n1 d|n d|n1 α α α α 1 2 k 1 k X 1 ns n∈A  Y 1 1+ s   ∞ ∞ p X |µ(n)| 1 X |µ(n)| Y 1 1 p = + = 1 + + ns 4s n=1 ns ps 4s 1 + 21s p n=1 2†n  = 1+ Y  1 1 ζ(s) 4s + 2s + 1 1 + = . 4s + 2s p ps ζ(2s) 4s + 2s ùÒ¤ ½n 2.7 y². y3y²½n 2.8. l$'¿d¼ê5ŸŒ±: X n≤x n∈A 1 = X n≤x |µ(n)| + X n≤ x4 2†n |µ(n)| = XX n≤x d2 |n µ(d) + XX n≤ x4 2†n µ(d) d2 |n 35 'uSmarandache¯KïÄ#? X = µ(d) + = = µ(d) 1≤l≤ dx2 X x = x µ(d) X X µ(d) d2 ≤ x4 1≤l≤ 4dx2 2†d 2†l 1 1≤l≤ 4dx2 d2 ≤ x4 2†d 2†l d ≤4 2†d d ≤x X µ(n) n2 6 = 2 +O π  n∈A = d ≤4 2†d d≤ 2 2†d    X  8 1 1 µ(n) , = 2 +O √ , x n2 π x √ x 6 +O 1 = x π2 n≤x X d ≤4 2†d X µ(d) x X µ(d) √ + + O( x), 2 2 d 8 √x d √ d≤ x u´k n≤ 2 2†n      √ x 8 1 1 √ + + O( + O x) x 8 π2 x √ 7 x + O( x). 2 π ùÒ¤ ½n 2.8 y². 36 X X  x  X  µ(d) 2 + O(1) + µ(d) + O(1) d 8d2 2 x d ≤x n≤x µ(d) + X X µ(d) X X µ(d) + O( |µ(d)|) + x + O( |µ(d)|) d2 8d2 2 2 x 2 x 2 = x Ù¥ 1+ X X d2 ≤x 1≤l≤ dx2 2†d2 l X d2 ≤x d2 ≤x µ(d) = d2 l≤ x4 d2 l≤x X X 1nÙ 'u Smarandache Ú¼ê˜ ¯K 1nÙ 'u Smarandache Ú¼ê˜ ¯ K 3.1 3.1.1 'u Smarandache Ú¼êþŠ Úó9(Ø é?¿ê n 9‰½ê k > 1, M. Bencze Q½Â ü ‡ Smarandache Ú¼ê S(n, k) 9 AS(n, k) Xe: S(n, k) = 9 X (n − ik) |n−ki|≤n i=0, 1, 2, ··· AS(n, k) = X |n−ki|≤n i=0, 1, 2, ··· |n − ik|. ~X, S(9, 4) = 9 + (9 − 4) + (9 − 8) + (9 − 12) + (9 − 16) = 5; S(11, 5) = 11 + (11 − 5) + (11 − 10) + (11 − 15) + (11 − 20) = 5; AS(9, 4) = 9 + |9 − 4| + |9 − 8| + |9 − 12| + |9 − 16| = 25; AS(11, 5) = 11 + |11 − 5| + |11 − 10| + |11 − 15| + |11 − 20| = 31. M. Bencze S(n, k) AS(n, k) , [32] [33]. , , . , , n k . , {S(n, k)} , n k , S(n, k) = 0? ? . , , . . , [8] [9]. S(n, k) AS(n, k) , . : Ӟ „ïÆ<‚ïÄ ¼ê 9 Žê5Ÿ ë©z 9 'uù˜ ¯K –8qvk<ïÄ –·‚vk3yk©z¥„ , ŠöǑùü‡¼ê´k¿Â –Œ±‡NÑê 3  ¥ ê Ú ÷v êꥩÙ5Ÿ d 3ê Ÿo^‡ž UÄǑxÑùaêAÆ ù Ñ´k¿ ÂïÄSN C ëïÄ ù˜¯K ¼ ˜ ? !ò 0 ù #óŠ ©¥¤9pd¼ê5Ÿ Œë©z 9 !̇8´|^{±9pd¼ê5Ÿïļ 9 þŠ5Ÿ ‰Ñü‡kþŠúª äN/` ê ǑÒ´y²e¡ü‡(Ø 37 ½n 3.1. ìCúª 'uSmarandache¯KïÄ#?  k > 1 Ǒ‰½ê, oé?¿¢ê x > 1, ·‚k 1 S(n, k) = 4 n≤x X  3 + (−1)k 1− 2k  x2 + R(x, k), Ù¥ |R(x, k)| ≤ 87 k + 5k8 x. ½n 3.2.  k > 1 Ǒ‰½ê, oé?¿¢ê x > 1, ·‚k ìCúª 2   7 + (−1)k 1 3 1 1+ x2 + R1 (x, k), x + AS(n, k) = 3k 4 2k n≤x X Ù¥ |R (x, k)| ≤ 87 k + 87 kx + 6kx + x2 . w,ùü‡ìCúª´'o÷, |^Ǒ)Ûêؐ{kŒ U°(ìCúª. 2 1 3.1.2 ½n 3.1 9½n 3.2 y² ù!·‚|^{±9pd¼ê5Ÿ†‰Ñ½ny ². Äk·‚^pd¼êò¼ê S(n, k) ?1{z, L«¤{ü /ª. 5¿ |n − ki| ≤ n  = −n ≤ n − ki ≤ n ½ö 0 ≤ i ≤  , Ù¥ [x] Ǒpd¼ê, =Ò´ [x] L«ØŒu x Œê. u´¼ ê S(n, k) ŒL«Ǒ: 2n k S(n, k) = 2n [X k ] (n − ik) = (n − ki) X |n−ki|≤n i=0, 1, 2, ··· = = 38  i=0       2n k 2n 2n n+n· − +1 k 2 k k    2  !   2 2n k 4n 2n 4n 2n 2n 2n + − + − − −n k 2 k k k k k k 1nÙ 'u Smarandache Ú¼ê˜ ¯K = 2n2 +n + k   2n k n − k 2  2n k 2 −  2n k ! , (3-1) Ù¥ {x} = x − [x] L« x ©êÜ©, 0 ≤ {x} < 1. y3é?¿ê x > 1,  x = k + r, ùp 0 ≤ r < k. u´ é (3-1) ªڌ: x k X n≤k [ = x k X n≤k [ x k S(n, k) ] n ] [ ] X x k =  2n k X =  2n k   k − 2 2n k (n + (i − 1)k) i=1 1≤n≤k = k − 2 n i=1 (i−1)k<n≤ik [ xk ] X X   2 2n k −   2n k  2n k !! 2 2n k 2n k 2 −  k − 2  !! −  2n k !! [ xk ] X    X 2n k 4n2 2n (n + (i − 1)k) − − k 2 k2 k k i=1 1≤n< 2 [ ] X    X 2n − k k (2n − k)2 2n − k + (n + (i − 1)k) − − k 2 k2 k i=1 k x k 2 <n<k [ ] X  X x k = i=1 1≤n<k = [ xk ] X k(k − 1) 2 i=1 = (n + (i − 1)k) (2i − 1) − k(k − 1) h x i2 2 k 2n k − k 2  2 4n 2n − 2 k k  − [ kx ] X X i=1 k ≤n<k 2 (ik − n) hxi X  X k h x i h x i 1+ n +1 2 k k k k k 2 ≤n<k 2 ≤n<k hxi X k hxi X k h x i2 X 1− 1+ n. (3-2) − 2 k k 2 k k k k 2 ≤n<k 2 ≤n<k 2 ≤n<k 39 'uSmarandache¯KïÄ#? u´d (3-2) ª 2|k žkðª: X S(n, k) = n≤k [ kx ] k 2 − 2k h x i2 k 2 − 2k h x i + . 4 k 8 k (3-3) k 2 − k h x i2 (k − 1)2 h x i + . 4 k 8 k (3-4)  k ǑÛêžkðª: X n≤k [ x k S(n, k) = ]  0 ≤ r ≤ k − 1 žk 0≤ = k[ X 1≤n≤r = X 1≤n≤r 2 5¿: ≤ (Üúª x k X S(n, k) ]<n≤k[ ]+r  h x i  2n  n+k − k k  h x i  2n  + n+k k k x k  !!  2n 2 2n − k k   2 ! k k 1 2n − − 8 2 k 2  k 2 5k k + x. 8 2 (3-5) x2 2x n x o n x o2 , + − k k2 k k k (3-2), (3-3) (3-5) k h x i2 9 X S(n, k) = n≤x = ·‚áǑ ǑóêžkìCúª: X n≤k [ = 1 4  x k S(n, k) + ] k[ 1− 2 k  x k X S(n, k) ]<n≤k[ ]+r x k x2 + R(x, k), Ù¥ |R(x, k)| ≤ 87 k + 5k8 x. (Üúª (3-2), (3-4) 9 (3-5) Œ k ǑÛêžkìCúª: 2 X n≤x 40 S(n, k) = X n≤k [ x k S(n, k) + ] k[ x k X S(n, k) ]<n≤k[ ]+r x k 1nÙ 'u Smarandache Ú¼ê˜ ¯K 1 = 4  1 1− k  x2 + R(x, k), Ù¥ |R(x, k)| ≤ 87 k + 5k8 x. u´y² ½n 3.1. y3·‚y²½n 3.2. d ¡OŽ S(n, k) y²{·‚k 2 AS(n, k) X = |n−ki|≤n i=0, 1, 2, ··· = [ nk ] X i=0 = = |n − ik| = |n − ki| + [ 2n k ] X 2n [X k ] i=0 |n − ki| |n − ki| ]+1   h i h i  k n h n i 2n n n+n − +1 −n − + k 2 k k k k       k 2n 2n k h n i h n i + +1 − +1 2 k k 2 k k   n n o2 2n n2 −k +n−n k k k n n o k  2n 2 k  2n  +k − + . (3-6) k 2 k 2 k i=[ n k hni ¤±d½n 3.1 (،: X AS(n, k) n≤x = = !   n n o k  2n 2 k  2n  n n o2 2n n2 +k − −k + +n−n k k k k 2 k 2 k n≤x !    2    X X  n2 2n k 2n k 2n n + − − +n − k k 2 k 2 k n≤x n≤x n n o X  n n o2 − k −k . k k n≤x X 5¿Oªµ 0 ≤ −k n n o2 k +k nno k ≤ k , 4 41 'uSmarandache¯KïÄ#? u´é?¿ê x,  k > 1 ǑêžkìCúªµ   7 + (−1)k 1 3 1 1+ x2 + R1 (x, k), x + AS(n, k) = 3k 4 2k n≤x X Ù¥ |R (x, k)| ≤ 87 k 1 3.2 3.2.1 2 7 x x + kx + + . 8 6k 2 u´¤ ½n 3.2 y². ˜ a  ¹ Smarandache Ú ¼ ê  Dirichlet ?ê Dirichlet S(n, k) ?ê†Ì‡(Ø !̇8´|^{±9¼ê5Ÿïļ ê S(n, k) Žâ5Ÿ±9˜a¹ S(n, k)  Dirichlet ?êOŽ ¯K. ¤¢ Dirichlet ?êÒ´/X X an , Ù¥ a ǑEê, s ǑEC þ. ù˜?ê3)ÛêØïÄ¥Ók›©­‡/ , NõͶêØ JKXxnâߎ!ƒê©Ù!iùbцƒ—ƒƒ', Ïdk ' Dirichlet ?ê5ŸïÄäk­‡nØ¿Â9Æâµ. !X ­0 ï² [34] (J, ǑÒ´é, AÏê k > 1, ‰Ñ? ê X S(n,n k) ˜‡äNOŽúª. äN/`ǑÒ´y²e¡A‡( Ø: ½n 3.3. é?¿Eê s Re(s) > 2, ·‚kðª ∞ n s n n=1 ∞ s n=1 ∞ X S(n, 4) n=1 ns     1 1 1 1 1 − s−1 ζ(s − 1) + 1 − s ζ(s), = 2 2 2 2 Ù¥ ζ(s) Ǒ Riemann zeta- ¼ê. ½n 3.4. é?¿Eê s Re(s) > 3, ·‚kðª ∞ X S(n2 , 8) n=1 42 ns 1 = 4     1 1 3 1 − s−2 ζ(s − 2) + 1 − s ζ(s). 2 4 2 1nÙ 'u Smarandache Ú¼ê˜ ¯K AO s = 4 ž, 5¿ ζ(2) = π6 , ζ(4) = π90 , ·‚kðª 4 2 ∞ X S(n2 , 8) n4 n=1 = π 2 5π 4 + . 32 604 ½n 3.5. é?¿Eê s ·‚kðª Re(s) > 3,       ∞ X 1 1 7 1 1 S(n2 , 16) = 1 + s−1 1 − s−2 ζ(s−2)+ + s−1 1 − s ζ(s). s n 2 2 8 2 2 n=1 AO s = 4 ž·‚kðª ∞ X S(n2 , 14) n4 n=1 = 9π 2 17π 4 + . 64 1440 ½n 3.6. é?¿Eê s ·‚kðª Re(s) > 3,     ∞ X S(n2 , 6) 1 1 1 2 1 − s−2 ζ(s − 2) + 1 − s ζ(s). = s n 3 3 3 3 n=1 AO s = 4 ž·‚kðª ∞ X S(n2 , 6) n4 n=1 = 4π 2 16π 4 + . 81 2187 ½n 3.7.  p > 2 Ǒƒê, Ké?¿Eê s ðª Re(s) > p, ·‚k
     ∞ X S np−1 , p 1 1 2 2 1 − s−p+1 ζ(s − p + 1) + 1 − 1 − s ζ(s). = ns p p p p n=1 3.2.2 A‡½ny² ù!·‚|^{±9pd¼ê5Ÿ†‰Ñù ½n y². Äk·‚^pd¼êò¼ê S(n, k) ?1{z, 5¿ (3-1) ªµ S(n, k) = n  2n k  k k + − 8 2  2n k  1 − 2 2 . (3-7) 43 'uSmarandache¯KïÄ#? Ù¥ {x} = x −] L« x ©êÜ©, 0 ≤ {x} < 1. ÏǑ 2nk = 2nk , XJ k > 2n. ¤± k > 2n žd (3-7) ªk𠪵   2n2 k 4n2 2n − − k 2 k2 k  2  ! 2n 2n − ≥ 0, k k S(n, k) = = n. qdu ¤±5¿ 0 ≤ k−1 , Šâ (3-7) ª·‚áǑíÑ k < 2n žkOª: k k − 2 0 ≤ S(n, k) ≤ n −  n k + . k 8 2n k  ≤ (3-8) d (3-8) ªØJíÑ Re(s) > 2 ž, Dirichlet ?ê F (s) = X S(n,n k) ýéÂñ, AOé k = 4, 5¿ k|2n žk S(n, k) = 0;  n ǑÛ êžk S(n, 4) = n +2 1 ; ζ(s) = X n1 = X (2n)1 + X (2n −1 1) = ∞ k s n=1 ∞ ∞ s n=1 ½ö ∞ s n=1 s n=1 ¤±·‚k   ∞ ∞ X X 1 1 1 1 ζ(s) + = 1 − s ζ(s), 2s (2n − 1)s (2n − 1)s 2 n=1 n=1 : ∞ ∞ ∞ X S(n, 4) X S(2n, 4) X S(2n − 1, 4) = + F4 (s) = ns 2s ns (2n − 1)s n=1 n=1 n=1 ðª = ∞ X S(2n − 1, 4) n=1 ∞ X (2n − 1)s = ∞ X n (2n − 1)s n=1 ∞ 1 1 1X 1 + = s−1 2 n=1 (2n − 1) 2 n=1 (2n − 1)s     1 1 1 1 = 1 − s−1 ζ(s − 1) + 1 − s ζ(s), 2 2 2 2 (3-9) Ù¥ ζ(s) Ǒ Riemann zeta- ¼ê. u´y² ½n 3.3. y3·‚y²½n 3.5. aq/Œ±íѽn 3.4.  Re(s) > 3 ž, d (3-7) ª·‚k ∞ X S(n2 , 16) n=1 44 ns 1nÙ 'u Smarandache Ú¼ê˜ ¯K 2 ∞ n X = n 2o n 8 − 16 2 ∞ X = n=1 + = ∞ X  = 1 8 1 8 n (2n−1)2 8 o 2s−2 (2n − 1)s−2 n 2 o n 8 − o − ∞ 8 X n (2n−1)2 2 o2 − n (2n−1)2 2 o 2s (2n − 1)s n o2 n o (2n−1)2 (2n−1)2 − ∞ 8 8 8 X n=1 − (2n − 1)s−2 n=1 (2n − 1)s X  X ∞ ∞ 1 1 1 7 1 + s−1 + + s−1 s−2 2 (2n − 1) 8 2 (2n − 1)s n=1 n=1      1 1 7 1 1 + s−1 1 − s−2 ζ(s − 2) + + 1 − s ζ(s). 2 2 8 2s−1 2 n=1  (2n−1)2 2 n 8 ns n=1 n n o 2 2 u´¤ ½n 3.5 y². k = 6 ž, 5¿ 3 ØØ    n žk n ≡ 1 (mod 3), ¤ 1 n ± 3 = 3 .  3 Ø n žk n3 = 0. ¤±d (3-7) ª·‚ k 2 2 2 ∞ X S(n2 , 6) n=1 2 = ∞ n X ns n 2o n 3 − = = = n (3n−1)2 3 n o 2 2 n 3 − ns n=1 ∞ X 6 2 o ∞ X n n 2 o n 3 (3n−1)2 3 o2 − n (3n−1)2 3 o −3 s−2 (3n − 1) (3n − 1)s n=1 n=1 n n o o2 n o (3n−2)2 (3n−2)2 (3n−2)2 ∞ ∞ − X X 3 3 3 + −3 s−2 s (3n − 2) (3n − 2) n=1 n=1 ! ! ∞ ∞ ∞ ∞ 2 X 1 1 X 1 1 X 1 1 X 1 + − − 3 n=1 ns−2 3s−2 n=1 ns−2 3 n=1 ns 3s n=1 ns     1 1 2 1 1 − s−2 ζ(s − 2) + 1 − s ζ(s). 3 3 3 3 45 'uSmarandache¯KïÄ#? u´y² ½n 3.6. Ǒy²½n 3.7, ·‚5¿é?¿ƒê p ≥ 3 ±9ê n (n,p) = 1,dͶ Euler(½ Fermat) ½n n ≡ 1 (mod p). u ´k 2np = 2p . u´dúª (3-7)  p−1 p−1 =
∞ X S np−1 , p ns n=1 n o o2 n p−1 o n p 2np−1 p−1 2np−1 − 2np −2 ∞ n p p X ns n=1  X  ∞ ∞ 1 1 2 2 X + 1− . = s−p+1 p n=1 n p ns n=1 (n, p)=1 du ∞ X n=1 (n, p)=1 1 = ns  1 1− s p (n, p)=1  ζ(s), dþª·‚áǑíÑðªµ
     ∞ X S np−1 , p 1 1 2 2 1 − s−p+1 ζ(s − p + 1) + 1 − 1 − s ζ(s). = ns p p p p n=1 u´¤ ¤k½ny². 3.3 3.3.1 ˜ a  ¹ Smarandache Ú ¼ ê  Dirichlet ?ê AS(n, k) ̇(Ø þ˜!0 ˜a¹ S(n, k)  Dirichlet ?êOŽ¯K, ! òUY0 'u AS(n, k)  Dirichlet ?êOŽ¯KïĤJ, d©Ù®ÜHŒÆƹ^. |^{±9¼ê5Ÿ ïÄ ˜a¹ AS(n, k)  Dirichlet ?êOŽ¯K, ¿é, AÏ 46 1nÙ 'u Smarandache Ú¼ê˜ ¯K ê k > 1, ‰ÑT?ꘇäNOŽúª. äN/`ǑÒ´y² e¡A‡(Ø: ½n 3.8. é?¿Eê s Re(s) > 3, ·‚kðª ∞ X AS(n, 2) n=1 ns 1 = 2ζ(s − 2) + 2s−1 ζ(s − 1), Ù¥ ζ(s) Ǒ Riemann zeta– ¼ê. ½n 3.9. é?¿Eê s Re(s) > 5, ·‚kðª ∞ X AS(n2 , 8) ns n=1      1 3 1 1 1 1 = ζ(s−4)+ + ζ(s−2)+ + 1 − s ζ(s). 8 4 2s 8 2s−1 2 π AO s = 6 ž, 5¿ ζ(2) = π6 , ζ(4) = π90 , ζ(6) = 945 , ·‚kð ª X 2 4 6 ∞ π 2 49π 4 7π 6 AS(n2 , 8) = + + . 4 n 48 5760 43008 n=1 ½n 3.10. é?¿Eê s ∞ X AS(n2 , 4) n=1 ns 1 = ζ(s − 4) + 4 AO s = 6 ž·‚kðª Re(s) > 5,  ∞ X AS(n2 , 4) n4 n=1 1 1 + s−1 2 2 = ½n 3.11. é?¿Eê s ∞ X AS(2n2 , 6) n=1 ns 2 = ζ(s − 4) + 3 AO s = 6 ž·‚kðª ∞ X AS(2n2 , 6) n=1 n4    1 1 1 − s ζ(s). ζ(s − 2) + 4 2 π 2 17π 4 π6 + + . 24 2880 3840 Re(s) > 5,  ·‚kðª 4 2 + s−1 3 3  ·‚kðª   1 2 1 − s ζ(s). ζ(s − 2) + 3 3 π2 83π 4 1456π 6 = + + . 9 10935 2066751 47 ½n 3.12. ðª 'uSmarandache¯KïÄ#?  p > 2 Ǒƒê. Ké?¿Eê s Re(s) > p, ·‚k ∞ X AS(np−1 , p) ns n=1   1 1 = 1 − s−2p+2 ζ(s − 2p + 2) p p      1 1 1 2 1 − s−p+1 ζ(s − p + 1) + 1 − s ζ(s). + 1− p p p p 3.3.2 ½ny² ·‚|^{±9pd¼ê5Ÿ†‰ÑA‡½n y². Äk·‚^pd¼êò¼ê AS(n, k) ?1{z, L«¤ {ü/ª . 5¿ |n − ki| ≤ n  = −n ≤ n − ki ≤ n ½   ö 0 ≤ i ≤ , Ù¥ [x] Ǒpd¼ê, =Ò´ [x] L«ØŒu x  Œê. u´¼ê AS(n, k) ŒL«Ǒ: 2n k AS(n, k) = X |n−ki|≤n i=0, 1, 2, ··· |n − ik| = [ 2n k ] X i=0 |n − ki| = [ nk ] X i=0 |n − ki| + 2n [X k ] ]+1 i=[ n k |n − ki|   h i  k h n i h n i 2n n = n+n· − +1 −n − k 2 k k k k      h i h i  2n k n n k 2n +1 − +1 + 2 k k 2 k k   n n o k  2n 2 k  2n  n n o2 n2 2n = +k − −k + , +n−n k k k k 2 k 2 k (3-10) hni Ù¥ {x} = x −] L« x ©êÜ©, 0 ≤ {x} < 1. ÏǑ 2nk = 2nk , XJ k > 2n. ¤± k > 2n žd (3-10) ªkð ª: AS(n, k) = 48 n2 2n2 n2 2n2 +n− − +n+ − n = n. k k k k 1nÙ 'u Smarandache Ú¼ê˜ ¯K ! − ≥ 0, ¤±5¿ 0 ≤ qdu k−1 , Šâ (3-10) ª·‚áǑíÑ k < 2n žkOª: k  k − 2 2n k  2 2n k  2n k n2 + n. 0 ≤ AS(n, k) ≤ k d  ≤ (3-11) ªØJ íÑ Re(s) > 3 ž§Dirichlet ?ê F (s) = ýéÂñ, AOé k = 2, 5¿ 2|n žk AS(n, 2) = X 1 1 n n + n;  n ǑÛêžk AS(n, 2) = + n + ; ζ(s) = = 2 2 2 n X 1 X X X 1 1 1 1 ½ö + = ζ(s) + = (2n) (2n − 1) 2 (2n − 1) (2n − 1)   1 ζ(s), ¤±·‚kðª: 1− 2 (3-11) AS(n, k) ns n=1 k ∞ X ∞ 2 2 s ∞ ∞ s n=1 ∞ ∞ s s n=1 s n=1 s n=1 n=1 s F2 (s) = = ∞ X AS(n, 2) ns n=1 ∞ (2n)2 X 2 n=1 = (2n)s n=1 ∞ (2n−1)2 X 2 + 2n + (2n)s n=1 ∞ X 2n2 n=1 = ∞ X AS(2n, 2) ns + ∞ X + ∞ X AS(2n − 1, 2) n=1 (2n − 1)s + 2n − 1 + (2n − 1)s 1 2 1 (2n)s−1 n=1 = 2ζ(s − 2) + 1 2s−1 ζ(s − 1), Ù¥ ζ(s) Ǒ Riemann zeta- ¼ê. u´y² ½n 3.8. y3·‚y²½n 3.10. aq/Œ±íѽn 3.9.  Re(s) > 3 ž, d (3-10) ª·‚k ∞ X AS(n2 , 4) n=1 = 4 ∞ n X 4 n=1 ns 2 2 +n −n n n2 2 o −4 n 2 o2 n 4 +4 ns n 2o n 4 +2 n 2 o2 n 2 −2 n 2o n 2 49 'uSmarandache¯KïÄ#? = ∞ 1 1X 1 ζ(s − 4) + ζ(s − 2) − 4 2 n=1 (2n − 1)s−2 n n o2 n o o2 n o (2n−1)2 (2n−1)2 (2n−1)2 (2n−1)2 ∞ ∞ − − X X 4 4 2 2 −4 +2 s s (2n − 1) (2n − 1) n=1 n=1 ∞ ∞ 1 1 1X 1X 1 ζ(s − 4) + ζ(s − 2) − + = s−2 4 2 n=1 (2n − 1) 4 n=1 (2n − 1)s     1 1 1 1 1 1 − s ζ(s). ζ(s − 4) + + ζ(s − 2) + = 4 2 2s−1 4 2 u´¤ ½n 3.10 y². k = 6 ž, 5¿ 3 ØØ    n žk n ≡ 1 (mod 3), ¤ ± n3 = 31 .  3 Ø n žk n3 = 0. ¤±d (3-10) ª·‚ k 2 2 2 ∞ X AS(2n2 , 6) n=1 = ns 2 2 ∞ (2n ) X 6 n=1 2 2 + 2n − 2n n 2n2 3 o ∞ X −6 2 n n n2 3 o2 +6 ns 2(3n−1)2 3 o n 2o n 3 ∞ X +3 n n o2 −3 o2 n 2n2 3 (3n−1)2 3 − n 50 o (3n−1)2 3 2 ζ(s − 4) + 2ζ(s − 2) − −6 3 (3n − 1)s−2 (3n − 1)s n=1 n=1 o2 n o o n n 2(3n−1)2 2(3n−1)2 2(3n−2)2 ∞ ∞ − 2 X X 3 3 3 +3 − (3n − 1)s (3n − 2)s−2 n=1 n=1 o2 n o o2 n o n n (3n−2)2 2(3n−2)2 (3n−2)2 2(3n−2)2 ∞ ∞ − − X X 3 3 3 3 −6 + 3 (3n − 2)s (3n − 2)s n=1 n=1 ! ∞ ∞ 4 X 1 1 X 1 2 ζ(s − 4) + 2ζ(s − 2) − − = 3 3 n=1 ns−2 3s−2 n=1 ns−2 ! ∞ ∞ 1 X 1 2 X 1 − + 3 n=1 ns 3s n=1 ns     1 2 4 2 2 1 − s ζ(s). ζ(s − 4) + + ζ(s − 2) + = 3 3 3s−1 3 3 = 2n2 3 o 1nÙ 'u Smarandache Ú¼ê˜ ¯K u´y² ½n 3.11. Ǒy²½n 3.12, ·‚5¿é?¿ƒê p ≥ 3 ±9ê n (n,p) = 1,dͶ Euler(½ Fermat) ½n n ≡ 1 (mod p). u ´k 2np = 2p . u´dúª (3-10)  p−1 p−1 ∞ X AS(np−1 , p) n=1 = ∞ X n=1 p 2 + ns  n2p−2  p  n 2np−1 p + np−1 − np−1 o2 − p 2 ns n 2np−1 p n 2np−1 p o ns −p n np−1 p o2 +p n np−1 p o o    X  ∞ ∞ ∞ 1 1 2 1 X 1 1 X + 1 − + . = s−p+1 s p n=1 ns−2p+2 p n p n n=1 n=1 (n,p)=1 du ∞ X n=1 (n, p)=1 1 = ns (n,p)=1 (n,p)=1  1 1− s p  ζ(s), dþª·‚áǑíÑðª: ∞ X AS(np−1 , p) ns   1 1 1 − s−2p+2 ζ(s − 2p + 2) = p p      2 1 1 1 + 1− 1 − s−p+1 ζ(s − p + 1) + 1 − s ζ(s). p p p p n=1 u´¤ ܽny². 'u Smarandache ˜ÚþŠ 3.4 3.4.1 ïĵ9̇(Ø 3 A!¥, ·‚0 Smarandache Ú¼ê S(n, k) 9 AS(n, k), 51 'uSmarandache¯KïÄ#? ¿‰Ñ A‡þŠúª±9¹ S(n, k)  Dirichlet ?ê. !¥·‚ ½Âü‡# Smarandache ¼ê, ¡Ǒ Smarandache ˜Ú¼ê P (n, k) 9 AP (n, k) Xe: P (n, k) = X (n − k i ) |n−k i |≤n i=0, 1, 2, ··· 9 AP (n, k) = X |n−k i |≤n i=0, 1, 2, ··· |n − k i |. ~X, P (12, 2) = (12 − 1) + (12 − 2) + (12 − 4) + (12 − 8) + (12 − 16) = 29, P (9, 4) = (9−1)+(9−4)+(9−16) = 6; P (11, 5) = (11−1)+(11−5) = 16; AP (12, 2) = |12−1|+|12−2|+|12−4|+|12−8|+|12−16| = 37; AP (9, 4) = |9 − 1| + |9 − 4| + |9 − 16| = 20; AP (11, 5) = |11 − 1| + |11 − 5| = 16. , , . , , n k . , P (n, k) AP (n, k) , . : ' uùü‡¼êˆ«Žê5Ÿ ·‚q˜ ¤ –vk3yk© z¥„ , ·‚Ǒùü‡¼ê´k¿Â –Œ±‡NÑ ê 3 ˜ê¥©Ù5Ÿ !ÄuÜ>óŠ |^ {±9pd¼ê5Ÿïļê 9 þŠ5Ÿ ‰Ñü‡kþŠúª äN/`ǑÒ´y²e¡ü‡(Ø ½n 3.13.  k > 1 Ǒ‰½ê, oé?¿¢ê x > 1, ·‚kì Cúª X P (n, k) = n≤x x2 · ln x + R(x, k), 2 ln k + 2x . Ù¥ |R(x, k)| ≤ x4 + x2 lnlnk2 + k −1 1 ·x− k x− 1 + 2x ln(2x)2+lnln(2x) k 2 2 2 ½n 3.14.  k > 1 Ǒ‰½ê, oé?¿¢ê x > 1, ·‚kì Cúª X AP (n, k) = n≤x Ù¥ |R (x, k)| ≤ 2x 1 52 2 + x2 · ln x + R1(x, k), 2 ln k x x2 (1 + ln 4) − . k−1 4 ln k 1nÙ 'u Smarandache Ú¼ê˜ ¯K le¡y²L§ǑØJwÑùü‡½nO¢Sþ´éo÷ , ´Ä3°(ìCúªǑ´˜‡k¯K. ½n 3.13 9½n 3.14 y² 3.4.2 ù!·‚|^{±9pd¼ê5Ÿ†‰Ñ½n y². Äk·‚^pd¼êò¼ê P (n, k) ?1{z, L«¤ |n − k | ≤ n  = −n ≤ n − k ≤ n ½ {ü/ª h . 5¿ i ö0≤i≤ , Ù¥ [x] Ǒpd¼ê, =Ò´ [x] L«ØŒu x Œê. u´¼ê P (n, k) ŒL«Ǒ: i i ln(2n) ln k P (n, k) [ ln(2n) ln k ] X i (n − k ) = (n − k i ) X = i=0 |n−k i |≤n i=0, 1, 2, ··· ln(2n)  k [ ln k ]+1 − 1 ln(2n) − = n+n· ln k k−1 ln(2n)   n · ln(2n) 2nk · k −{ ln k } − 1 ln(2n) = − +n−n ln k ln k k−1 ln(2n)   n · ln(2n) 2nk · k −{ ln k } 1 ln(2n) − = +n+ −n ,(3-12) ln k k−1 ln k k−1  Ù¥ {x} = x − [x] L« x ©êÜ©, 0 ≤ {x} < 1. y3é?¿ê x > 1, d Euler Úúª (ë©z [8] ¥½n 4.9) Œ: ½ö Z x 1 X n · ln(2n) y · ln(2y) dy ≤ ≤ ln k ln k n≤x Z x+1 1 y · ln(2y) dy, ln k X n · ln(2n) x2 x2 · ln(2x) − ≤ 2 ln k 4 ln k n≤x ≤ (x + 1)2 (x + 1)2 · ln(2x + 2) − . (3-13) 2 ln k 4 53 'uSmarandache¯KïÄ#? u´d (3-13) ªáǑOª: X n · ln(2n) ln k n≤x − 2x ln(2x) + ln(2x) + 2x x2 x2 ≤ · ln(2x) + .(3-14) 2 ln k 4 2 ln k u´(Ü (3-12) 9 (3-14) ªŒ X P (n, k) n≤x = X n≤x = X n · ln(2n) n≤x 2 = ! ln(2n)  2nk · k −{ ln k } ln(2n) − ln k k−1 ! ln(2n)   2nk · k −{ ln k } ln(2n) 1 − +n−n k−1 ln k k−1 n · ln(2n) 1 +n+ −n ln k k−1 + ln k X n≤x  x · ln(2x) + R(x, k), 2 ln k + 2x . u´ Ù¥ |R(x, k)| ≤ x4 + k −1 1 · x − k x− 1 + 2x ln(2x)2+lnln(2x) k y² ½n 3.13. y3·‚y²½n 3.14. †y²½n 3.13 aq, ·‚ké AP (n, k) ?1{z. d AP (n, k) ½Â·‚k 2 2 AP (n, k) = X |n−k i |≤n i=0, 1, 2, ··· = ln n [X ln k ] i=0 n − ki = (n − k i ) + [ ln(2n) ln k ] X i=0 n − ki [ ln(2n) ln k ] X (k i − n) n i=1+[ ln ln k ] ln n [X [ ln(2n) ln k ] ln k ] X i = 2 (n − k ) + (k i − n) i=0 i=0   ln n k [ ln k ]+1 − 1 ln(2n) ln n −2· −n−n· = 2n + 2n · ln k k−1 ln k ln(2n) +1 k [ ln k ] − 1 + k−1  54  1nÙ 'u Smarandache Ú¼ê˜ ¯K     ln n 1 ln(2n) n · ln(n/2) − 2n +n+ +n = ln k k−1 ln k ln k   ln(2n) ln n 2nk (3-15) + · k −{ ln k } − k −{ ln k } . k−1 u´d (3-15) ª9 Euler ÚúªáǑOª X AP (n, k) n≤x = X  n · ln(n/2) n≤x + = Ù¥ ². ln k 1 +n+ +n k−1  X 2nk  ln(2n) ln n · k −{ ln k } − k −{ ln k } k−1 n≤x x2 · ln x + R1 (x, k), 2 ln k ln(2n) ln k  x2 (1 + ln 4) x − . |R1 (x, k)| ≤ 2x + k−1 4 ln k 2  − 2n  ln n ln k  u´¤ ½n 3.14 y 55 'uSmarandache¯KïÄ#? 1oÙ 'uŒ\¼ê˜ ¯K ˜‡#Œ\¼ê† Smarandache ê 4.1 4.1.1 Úó9(Ø é?¿ê n, ·‚¡Žâ¼ê f (n) ´Œ\, XJé?¿ ê m, n (m, n) = 1 k f (mn) = f (m) + f (n). ¡ f (n) ´Œ\, XJé?¿ê r 9 s Ñk f (rs) = f (r) + f (s). ½Â 4.1. ·‚½Â˜‡#Žê¼ê F (n) Xe: F (0) = 0,  n > 1 n IO©)ªǑ n = p p · · · p ž, ½Â F (n) = α1 α2 1 2 α1 p1 + α2 p2 + · · · + αk pk . ¯¢þ m = p 9 αk k ž ·‚k l d ½Âk ¤± ´˜‡ Œ\¼ê 3êØ¥ ÷vŒ\5ŸŽâ¼êéõ ~X I O©)ªǑ ž ¼ê 9é ê¼ê Ñ´Œ\¼ê d ê ¤kØÓƒÏ ê‡ê ´˜‡Œ\¼ê Ø´Œ\¼ê 'uŒ\¼ ê5ŸïÄ 3êر9ƒê©Ù¯K¥Ók›©­‡ ˜ N õͶêØJKцƒ—ƒƒ' Ï ÙïÄóŠ´ék¿Â k '¼ê 9 5Ÿ Œë©z 9 !̇8´ïÄŒ\¼ê 3, AÏêþþ Š©Ù¯K ¿|^{‰ü‡rìCúª Ǒd ·‚k0 ê 9 3©z 9 ¥ {7 ˜a ÛêZæêØ;[ ÇÚ\ Nõêؼê9ê ¿ JÑ Ø™)û¯K 9 Ò´Ù¥ü‡k ê L« ¤kÏêÈ L« ¤ku êÏfÈ =Ò´ · · · pαk k n = pβ1 1 pβ2 2 · · · pβk k , mn = F (n) F (mn) = (α1 + β1 )p1 + (α2 + β2 )p2 + · · · + (αk + βk )pk = F (m) + F (n). F (n) . , , n αk α1 α2 n = p1 p2 · · · pk , Ω(n) = α1 + α2 + · · · + αk f (n) = ln n . , n ω(n) = k , . , , , . Ω(n) ω(n) , [35] [36]. F (n) , . , Smarandache {Pd (n)} {qd (n)}. [1] [37] , F. Smarandache , , {Pd (n)} {qd (n)} , Pd (n) n , qd (n) n n . Y Y d(n) d(n) Pd (n) = d=n 2 ; qd (n) = d = n 2 −1 , α1 α2 1 p2 α1 +β1 α2 +β2 αk +βk , p1 p2 · · · pk d|n 56 d|n,d<n 1oÙ 'uŒ\¼ê˜ ¯K Ù¥ d(n) Ǒ Dirichlet Øê¼ê, = n ¤kÏê‡ê. 3©z [1] ¥, F. Smarandache ÇïÆ·‚ïÄê {P (n)} 9 {q (n)} 5Ÿ. 'uù˜¯K, NõÆö?1LïÄ, ¼ ˜X k(Ø, ë©z [38] 9 [39]. !|^9)ې{ïÄ ¼ ê F (n) 3ê {P (n)} 9 {q (n)} þþŠ¯K, ¿‰Ñ ü‡r þŠúª. äN/`ǑÒ´y² e¡: ½n 4.1.  N Ǒ‰½ê, Ké?¿‰½¢ê x > 1, ·‚ kìCúª d d d d N X x2 di · i + O F (Pd (n)) = ln x i=1 n≤x X  x2 lnN+1 x  , Ù¥ d (i = 1, 2, · · · , N) ǑŒOŽ~ê d = π72 . ½n 4.2.  N Ǒ‰½ê, Ké?¿‰½¢ê x > 1, ·‚ kìCúª 4 i 1 N X x2 hi · i + O F (qd (n)) = ln x i=1 n≤x X Ù¥ h i (i = 1, 2, · · · , N)  ǑŒOŽ~ê x2 lnN+1 x  , π4 π2 h1 = − . 72 12 ü‡{üÚn 4.1.2 Ǒ ¤½ny², ·‚I‡Xe{üÚn: Ún 4.1.1. é?¿¢ê x > 1,  π(x) L«¤k،u x ƒê ‡ê, Ké?¿ê k, ·‚kìCúª: k X x π(x) = 1= ci · i + O ln x i=1 p≤x X  Ù¥ c (i = 1, 2, · · · , k) ǑŒOŽ~ê c y²: ë©z [27] ¥1nÙ½n 2© i x lnk+1 x 1  , = 1. 57 'uSmarandache¯KïÄ#?  N Ǒ‰½ê, Ké?¿‰½¢ê x > 1, · Ún 4.1.2. ‚kìCúª N X x2 F (n) = di · i + O ln x i=1 n≤x X  x2 lnN+1 x  , Ù¥ d (i = 1, 2, · · · , N) ǑŒOŽ~ê d = π12 . y²: é?¿ê n,  P (n) L« n ŒƒÏf. y3·‚½ ÂXeü‡8Ü: 2 i 1 A = {n : n ≤ x, P (n) ≤ √ n}; B = {n : n ≤ x, P (n) > √ n}.  n ∈ A ž, d F (n) ½ÂN´íÑ F (n) ≪ √n ln n. u´Š â Abel ð (ë©z [8] ½n 4.2) ·‚Œ: X n≤x n∈A ·‚k F (n) ≪ X√ n≤x 3 n ln n ≪ x 2 ln x. éu8Ü B, 5¿ F (n) Œ\5Ÿ, dÚn 4.1.1 9 Abel ð N  x  x Z xk X x2 p=π · − π(y)dy = ri · 2 i k k k ln 1 i=1 p≤ x X k Ù¥ r (i = 1, 2, · · · , N) ǑŒOŽ~ê u´|^þª·‚ØJíÑ X F (n) = X X √ x k≤ x k<p≤ k 58 "N X X √ k≤ x  p+O x2 ri · 2 i k ln i=1 X F (n) = n≤x √ p|n, p> n n∈B = X F (pk) = √ x k≤ x k<p≤ k +O  k = x2 k 2 lnN+1 +O x k ! , X X (F (k) + p) √ x k≤ x k<p≤ k pk≤x p>k X X x k x k x2 k 2 lnN+1 1 r1 = . 2 i = (4-1) X X √ k≤ x x k !# k<p≤ kx 3 p+O x2 ln x ! 1oÙ 'uŒ\¼ê˜ ¯K N X x2 = di · i + O ln x i=1  x2 lnN+1 x  , (4-2) Ù¥ d (i = 1, 2, · · · , N) ǑŒOŽ~ê (Ü (4-1) 9 (4-2) áǑìCúª: 1 π2 d1 = ζ(2) = . 2 12 i N X x2 di · i + O F (n) = F (n)) + F (n) = ln x i=1 n≤x n≤x n≤x X X X n∈B n∈A u´y² Ún 4.1.2.  x2 lnN+1 x  . ½n 4.1 9½n 4.2 y² 4.1.3 ù!·‚|^{±9© n؉ѽn 4.1 9½n 4.2 y ². ·‚Äky²½n 4.1. dê {P (n)} ½Â¿5¿© ð (ë ©z [8] ¥½n 3.17) ·‚k d X F (Pd (n)) n≤x = X  F n d(n) 2  = X1 d(n)F (n) 2 X 1 X 1 X F (mn) = (F (m) + F (n)) = F (n) = 2 2 nm≤x nm≤x mn≤x     X X X X X X F (m) ·  1 . F (n) + F (n) −  = n≤x n≤x √ x n≤ x m≤ n √ x m≤ x n≤ m √ n≤ x √ m≤ x (4-3) dÚn 4.1.2 ·‚k X X √ m≤ x x n≤ m F (n) = X √ m≤ x N X "N X x2 di · 2 i m ln i=1 x2 ui · i + O = ln x i=1  x m +O x2 lnN+1 x   , x2 m2 lnN+1 x # (4-4) 59 'uSmarandache¯KïÄ#? Ù¥ u (i = 1, 2, · · · , N) ǑŒOŽ~ê u A^ Abel ð9Ún 4.1.2 ·‚kOª i 1 = d1 · ζ(2) =   X X F (n) +O F (n) n √ √ n≤ x n≤ x ! 3 x2 . = O ln x X X F (n) = x · √ x n≤ x m≤ n π4 . 72 (4-5) Ów,ǑkOª   X √ m≤ x   F (n) ·  X √ n≤ x  3 x2 1 ≪ . ln x (4-6) (Ü (4-3), (4-4), (4-5) 9 (4-6) áǑíÑìCúª: N X x2 F (Pd (n)) = ui · i + O ln x i=1 n≤x X  x2 lnN+1 x  , Ù¥ u (i = 1, 2, · · · , N) ǑŒOŽ~ê u = π72 . u´y² ½ n 4.1. 5¿½n 4.1 ¿A^Ӑ{·‚ǑŒ± 4 i X 1 X  d(n)  X 1X d(n)F (n) − F (n) 2 2 n≤x n≤x n≤x   N N X X x2 x2 x2 di · i + O ui · i − = ln x ln x lnN+1 x i=1 i=1   N X x2 x2 , = hi · i + O ln x lnN+1 x i=1 F (qd (n)) = n≤x − 1 F (n) = Ù¥ h = u − d (i = 1, 2, · · · , N) ǑŒOŽ~ê u´¤ ½n 4.2 y². i 60 i i h1 = π4 π2 − . 72 12 1oÙ 'uŒ\¼ê˜ ¯K 4.2 'uŒ\¼êþŠ 4.2.1 ̇(Ø þ˜!·‚ïÄ ¼ê F (n) 3˜ AÏêþþŠ5Ÿ, ¼ ü‡rìCúª. ŠǑù˜óŠò, !·‚ļê F (n) Š©Ù5Ÿ. =Ò´ïÄþŠ (F (n) − P (n)) þŠ5Ÿ, ¿|^ {±9ƒê©ÙnØ‰Ñ (F (n) − P (n)) ˜‡rþŠú ª. äN/`ǑÒ´y²e¡: ½n 4.3.  N Ǒ‰½ê, Ké?¿‰½¢ê x > 1, ·‚ kìCúª 2 2 X n≤x 2 (F (n) − P (n)) = N X i=1 ci · x2 lni+1 x +O  x2 lnN+2 √ x  , Ù¥ c (i = 1, 2, · · · , N) ǑŒOŽ~ê c = π6 . ½n 4.3 ¿Â3u§U `²¼ê F (n) ŠÌ‡8¥3 ê n ŒƒÏfþ. Ïdù &E‰·‚Jø ˜‡››¼ê F (n) Œ­‡å». 2 i 1 ½n 4.3 y² 4.2.2 ù!·‚|^{±9ƒê©Ùn؆‰Ñ½n 4.3 y ². ·‚æ^©z [3] ¥gŽ. Äk½Âo‡8Ü A, B, C, D Xe: A = {n, n ∈ N, n Tk˜‡ƒÏf p ÷v n = kp, p > n , k ¤kƒ Ïf q ÷v q < n }; B = {n, n ∈ N, n k˜‡ƒÏf p ÷v n = p · k, p > n > k}; C = {n, n ∈ N, n kü‡ƒÏf p 9 p ÷v n = p p k, p > p > n > k}; D = {n, n ∈ N, n ¤kƒÏf p ÷v p ≤ n }, Ù ¥ N L«¤kêƒ8. u´d8Ü A, B, C 9 D ½ÂŒ 1 3 1 3 2 1 3 2 X n≤x 1 1 (F (n) − P (n))2 = 2 1 2 1 3 1 3 X n≤x n∈A (F (n) − P (n))2 + X n≤x n∈B (F (n) − P (n))2 61 'uSmarandache¯KïÄ#? + X n≤x n∈C (F (n) − P (n))2 + X n≤x n∈D (F (n) − P (n))2 ≡ W1 + W2 + W3 + W4 . (4-7) y3·‚|^{±9ƒê©ÙnØ5O (4-7) ¥ˆ‘. Äk ·‚O W . 5¿ F (n) ǑŒ\¼ê  n ∈ A n = pk, k  ¤kƒÏf q ÷v q ≤ n ž, F (k) ≤ n ln n ±9ƒê©Ù½n (ë ©z [27] ¥1nÙ½n 2) 1 1 3 Ù¥ c k X x π(x) = 1= ci · i + O ln x i=1 p≤x X ǑŒOŽ~ê  x lnk+1 x (i = 1, 2, · · · , k) c1 = 1. X X = (F (n) − P (n))2 = (F (pk) − p)2 i W1 1 3 n≤x n∈A =  , (4-8) ·‚kOª: pk≤x (pk)∈A X pk≤x (pk)∈A F 2 (k) ≪ 2 ≪ (ln x) √ x k≤ x k<p≤ k k≤ x X n≤x n∈B (F (n) − P (n))2 = X 1 X k≤x 3 k<p≤ ≪ W4 = X 2 k3 √ k≤ x X X 1 k≤x 3 X n≤x n∈D x 3 2 √x k ln x ≪
2 F (p2 k) − p p2 k≤x p>k k 3 2 (4-9) X (F (k) + p)2 ≪ X 1 X k≤x 3 k<p≤ √x p2 k 3 2 x . ln x (F (n) − P (n))2 ≪ (4-10) X n≤x 2 5 n 3 ln2 n ≪ x 3 ln2 x. (4-11)  ·‚Ȏ W . 5¿ n ∈ C ž, n = p p k, Ù¥ p > p n > k. XJ k < p < n , ù«œ¹áu W O. XJ k < p 3 1 3 62 1 1 2 1 3 2 p3 k<p≤ xk 2 3 √ = 2 (pk) 3 ln2 (pk) ≤ (ln x)2 x5 1 5 3 2 3 k x ≪ x ln x. k ln k X W2 = X X 1 2 1 1 > < p2 1oÙ 'uŒ\¼ê˜ ¯K < n , ù«œ¹áu W O. u´A^ (4-8) ª·‚k W3 1 3 4 = X (F (n) − P (n)) = n≤x n∈C = X X X X 1 k≤x 3 k<p1 ≤ = 1 k≤x 3 k<p1 ≤  √x p X √x p X 1 k≤x 3 k<p1 ≤ X X p1 p2 k≤x p2 >p1 >k X √x p k x 1 <p2 ≤ p k 1 N X √ k<p1 ≤ xk  5  X 2 3 +O x ln x − 1 k≤x 3 X 1 k≤x 3 k<p1 ≤  X +O  1 k≤x 3 5¿ ·‚k π2 ζ(2) = , 6 X 1 k≤x 3 = X 1 k<p1 ≤ = = X √x p k 2    kp1  x p1 k X √ x p2 ≤p1 k  x 1 <p2 ≤ p k 1 +O  x p1 k lnN+1 x ! p21  kp1  . (4-12) A^ Abel ð (ë©z [8] ¥½n 4.2) 9 (4-8) ª X k<p1 ≤ √x p21 √x X 1 p≤p1 k X k≤x 3 k<p1 ≤ N X X 5 3 (F (p1 p2 k) − p2 ) + O x ln x x ci · p1 k lni i=1 p21   5  p21 + O x 3 ln2 x x 1 <p2 ≤ p k 1 k 2  5 
F 2 (k) + 2p1 F (k) + p21 + O x 3 ln2 x X x 2≤ p k 1 k X +O  = X 2 p21 N X ci · p1 i i=1 k ln p1 +O  ln p1 N+1  p1 !  X ci · p31 p31  X + O   i N+1 ln p ln p √ √ 1 1 1 1 x x i=1 k≤x 3 k<p1 ≤ k k≤x 3 k<p1 ≤ k   N X X X ci · p3 p31  X X 1 + O   √ x lni p1 √ x lnN+1 p1 1 1 i=1 X X k≤x 3 p1 ≤ k k≤x 3 p1 ≤ k 63 'uSmarandache¯KïÄ#? N X X = i=1 3 ci x 2 p π 3 k 2 lni xk 1 k≤x 3  X +O  1 X k≤x 3 p1 ≤ i p31 √ x ln N+1 p1 k    N 2 N X di · x 2 2 ·x +O , i+1 N+2 ln x ln x i=1 = Ù¥ d  3 ! r  Z √ x k ci y x − π(y)d k lni y 2  ǑŒOŽ~ê (i = 1, 2, · · · , N) X 1 X k≤x 3 k<p1 ≤ √x p k X x 1 <p2 ≤ p k 1 kp1 ≪ 1 k≤x 3 X X k 1 p1 ≤ x X 1 k≤x 3 π2 . 6 √x p1 · x p1 k ln x k 3 2 5 x3 √ . ≪ ln2 x k ln2 x X x2 p1 x x2 ≪ ≪ . √ x k lnN+1 x k 2 lnN+2 x lnN+2 x 1 X k<p1 ≤ d1 = k≤x 3 ≪ X (4-13) (4-14) (4-15) k≤x 3 k ÓnA^ Abel ð, (4-8) ª±9 (4-13) y²{·‚ǑŒ±ì Cª X 1 X k≤x 3 k<p1 ≤ = X 1 k 1 k≤x 3 = N X i=1 ai · √x p21 x p1 k ln px1 k k xp1 x √ x ln kp1 X k<p1 ≤ x2 lni+1 x k +O  x2 lnN+2 x  , (4-16) Ù¥ a (i = 1, 2, · · · , N) ´ŒOŽ~ê a = π3 . u´(Ü (4-7), (4-9), (4-10), (4-11), (4-12), (4-13), (4-14), (4-15) 9 (4-16) ·‚áǑíÑìCúª: 2 i X n≤x 64 1 2 (F (n) − P (n))   N X x2 di · x 2 +O = ai · i+1 − i+1 N+2 √ ln x ln x ln x i=1 i=1 N X x2 1oÙ 'uŒ\¼ê˜ ¯K = N X i=1 hi · x2 lni+1 x +O  x2 lnN+2 √ x Ù¥ h = a − d (i = 1, 2, · · · , N) ǑŒOŽ~ê π π π − = . u´¤ ½n 4.3 y². 3 6 6 2 i 2 i 2 i  , h1 = a1 − d1 = 65 'uSmarandache¯KïÄ#? 1ÊÙ 'u Smarandache ê9Ùk'¯ K 5.1 Smarandache Š ²ê SP (n) Ú IP (n) þ ½Â 5.1. é?¿šKê n, ·‚^ SP (n) L« n  Smarandache ²ê, =Ò´Œu½u n ²ê. ~XTê A‘Ǒ: 0, 1, 4, 4, 4, 9, 9, 9, 9, 9, 16, 16, 16, 16, 16, 16, 16, 25, · · · . ½Â 5.2. ^ IP (n) L« n  Smarandache Œ²ê, =Ò´Ø ‡L n Œ²ê. ù‡ê A‘Ǒµ0, 1, 1, 1, 4, 4, 4, 4, 4, 9, 9, 9, 9, 9, 9, 9, 16, 16, 16, 16, 16, 16, 16, 16, 16, 25, · · · . Sn = (SP (1) + SP (2) + · · · + SP (n)) /n; In = (IP (1) + IP (2) + · · · + IP (n)) /n; p Kn = n SP (1) + SP (2) + · · · SP (n); p Ln = n IP (1) + IP (2) + · · · IP (n). 3©z [1] ¥, F. Smarandache ÇJÑ ùü‡ê, ¿ïÆ< ‚ïħˆ«5Ÿ, k'ù SNÚk'µë©z [1]![37]![40] 9 [41]. 3©z [40] ¥, F Kenichiro Kashihara Ƭ2géùü‡ê ) ,, ӞJÑ ïÄ4 SI !(S − I )! KL 9 (K − L ) ñÑ5¯K, XJÂñ, ¿Ù4. 3©z [41] ¥, ƒÄgïÄ ùA‡þŠìC5¯K, ¿|^9)ې{y² e¡A‡(Ø: ½n A. éu?¿¢ê x > 2, kìCúª n n X n6x X n6x 66 n  3 x2 SP (n) = + O x2 ; 2 IP (n) =  3 x2 + O x2 . 2 n n n n n 1ÊÙ 'u Smarandache ê9Ùk'¯K dd½náǑíÑe¡íØ: íØ 1. é?¿ê n, kìCª   S S =1+O n = 1. 94ª lim I I − 12 n n n→∞ n íØ 2. é?¿ê n, kìCª   K K =1+O n 94ª lim L L − 12 n n n→∞ n n n = 1, lim (Kn − Ln ) = 0. n→∞ , , 'u S − I ìC5¯K, qvk3©z [41] ¥9. , , ·‚Ǒù˜¯K´k, ÙÏ3u˜¡§)ûŒ±é© z [40] ¥¯KŠ±£‰, xþ ÷éÒ; ,˜¡„Œ±Ǒ xÑü«ê SP (n) 9 IP (n) Ÿ«O. !Qão®Ú [42] ó Š, Äu©z [41] ¥gŽ¿(ÜÓa‘Ü¿±9Ø ‘°(?n, ¨ïÄ S − I ìC5¯K, ¼ ˜‡rìCúª, äN/` ǑÒ´y² e¡: ½n 5.1. éu?¿ê n > 2§·‚kìCúª n n n n Sn − In = 4√ n + O (1) . 3 w,©¥(J‘Ö ©z [41] ¥Øv, Ӟò©z [40] ¥é ê S 9 I JѤk¯K‰ƒ )û. ,, dd½n·‚„Œ± íÑe¡4: S −I 4 lim (S − I ) = 1 9 lim √ = . n 3 n n n n→∞ n 1 n n n n→∞ y²: e¡·‚^{9 Euler Úúª (ë©z [8]) © Oé S 9 I ?1š~°(O, ª|^ü‡°(O‰Ñ½ n 5.1 y². é?¿ê n > 2, w,3˜ê M ÷v: M < n 6 (M + 1) , = M = n + O(1). u´k n n Sn = 1 2 2 2 1X 1 X 1 SP (k) = SP (k) + n n n 2 k6n k6M X SP (k) M 2 <k6n 67 'uSmarandache¯KïÄ#? = 1 X n X SP (k) + h6M (h−1)2 <k6h2 1 n X (M + 1)2 M 2 <k6n 
1 1 X 2 h − (h − 1)2 h2 + n − M 2 (M + 1)2 n n h6M
1
1 X 2h3 − h2 + n − M 2 (M + 1)2 = n n = h6M
M 2 (M + 1)2 M (M + 1) (2M + 1) 1 = n − M 2 (M + 1)2 − + 2n 6n n 2 M (M + 1) (2M + 1) M (M + 1)2 = (M + 1)2 − − . (5-1) 6n 2n Ón, Šâ IP (n) ½Â, ·‚ǑkOŽúª: In = = = = 1 X 1 X 1X IP (k) = IP (k) + IP (k) n n n 2 k6n k<M 2 M 6k6n X 1 X 1 X IP (k) + M2 n n h6M (h−1)2 6k<h2 M 2 6k6n 
1 1 X 2 h − (h − 1)2 (h − 1)2 + n − M2 + 1 M2 n n h6M
1
1 X 2h3 − 5h2 + 4h − 1 + n − M2 + 1 M2 n n h6M = = M 2 (M + 1)2 5M (M + 1) (2M + 1) − 2n 6n
n − M2 + 1 M2 2M (M + 1) M + − + n n n 2 2 M (M − 2M − 3) 5M (M + 1) (2M + 1) M2 − − 2n 6n 2M 2 + M + . n u´d (5-1) 9 (5-2) ªŒ Sn − In = (M + 1)2 − 68 M (M + 1) (2M + 1) M 2 (M + 1)2 − 6n 2n (5-2) 1ÊÙ 'u Smarandache ê9Ùk'¯K  M 2 (M 2 − 2M − 3) 5M (M + 1) (2M + 1) 2M 2 + M − + − M − 2n 6n n 3 2M + 7M = 2M + 1 − . (5-3) 3n  2 5¿ M = n 1 2 + O(1), Sn − In = 2M − d (5-3) ª·‚áǑíÑ 2M 4 4√ + O(1) = M + O(1) = n + O(1). 3 3 3 u´¤ ½n 5.1 y². Smarandache 3n-digital 5.2 ê ½Â 5.3. é?¿ê n, Ͷ Smarandache 3n-digital ê  {a } ½ÂǑ: {a } = {13, 26, 39, 412, 515, 618, 721, · · · }. =, é? ¿ê b ∈ {a }, §Œ±©Ǒü‡Ü©, Ù¥1Ü©´1˜Ü© 3 . ~X, a = 2884, a = 35105, a = 104312, · · · . 3©z [1] ¥ F. Smarandache ïÆ·‚ïÄê {a(n)} 5Ÿ. ' uù˜êéõÆö®²?1 ïÄ¿ ˜X¤J. 3©z [43] ¥, ©¡±Ú
[ ïÄ ù˜êþŠ¯K, ‰Ñ ±e½n: ½n 5.2. é?¿¢ê N > 1, kìCúª n n n 28 35 104 X n 3 = · ln N + O (1) . an 10 ln 10 n≤N y²: ·‚^{¤½ny². Äk, é?¿ ê n, - 3n = b b · · · b b , Ù¥ 1 ≤ b ≤ 9, 0 ≤ b ≤ 9, i = 1, 2, · · · , k(n) − 1. d a ½ÂŒ a ŒǑ: 2 1 k(n) k(n)−1 n u´k k(n) i n an = n · 10k(n) + 3n = n · (10k(n) + 3). X n X 1 . = k(n) an 10 + 3 n≤N n≤N 69 'uSmarandache¯KïÄ#? w,e N ≤ 3, K X n 3 − log10 N an 10 n≤N ´˜‡~ê. Ïd, ؔ˜„5, ·‚b N > 3. dž, 3ê M  · · 33} . 33 · · 33} < N ≤ 33 | ·{z | ·{z M M+1 5¿, é?¿ê n, e 33 · · 33} < n ≤ 33 · · 33}, | ·{z | ·{z u u−1 · · · b2 b1 . X n an n≤N X 1 n≤3 10 + 3 + = = X 3<n≤33 1 + 2 10 + 3 M X 33<n≤333 1 33 · · 33}<n≤33 · · 33} | ·{z | ·{z M−1 = u´k X + = u u−1 K 3n = b b = 10M +3 + 1 + ··· +3 X 103 33 · · 33}<n≤N | ·{z 1 10M+1 +3 M M 3 30 300 3 · 10M−1 N − 10 3 −1 + 2 + 3 + ··· + + M+1 10 + 3 10 + 3 10 + 3 10M + 3 10 +3   M 2 3 M N − 10 3 −1 3 10 10 10 10 + M+1 + + + ··· + M 10 10 + 3 102 + 3 103 + 3 10 + 3 10 +3       3 3 3 3 1− + 1− 2 + 1− 3 + ··· 10 10 + 3 10 + 3 10 + 3   M N − 10 3 −1 3 + 1− M + M+1 10 + 3 10 +3    3 3 3 3 3 M− + + + ··· + M 10 10 + 3 102 + 3 103 + 3 10 + 3 M N − 10 3 −1 + M+1 10 +3 M M N − 10 3 −1 3 1 9 X = ·M − · + M+1 10 10 i=1 10i + 3 10 +3 70 (5-4) 1ÊÙ 'u Smarandache ê9Ùk'¯K = 3 · M + O(1). 10 (5-5) y3·‚5O M , d (5-4) ªŒ 10M − 1 < 3N ≤ 10M+1 − 1     1 1 M ln 10 + ln 1 − M < ln(3N) ≤ (M + 1) ln 10 + ln 1 − M+1 10 10 1 ) ln(3N) ln(1 − 101M ) ln(3N) ln(1 − 10M+1 − −1≤M < − . ln 10 ln 10 ln 10 ln 10 1 N → +∞ , ln(1 − 10M+1 ) ∼ 101M , ln(1 − 101M ) ∼ 101M . 5¿ ž ln 3N −1−O ln 10 Ï  1 10M (ÜdªÚ (5-5) ª, áǑíÑ  ln 3N ≤M < −O ln 10  1 10M  , . X n 3 = · ln N + O (1) . an 10 ln 10 n≤N ùÒ¤ ½ny². 'uù˜ê, Ü©+DŽJÑ Xeߎ: ߎ. 3 Smarandache 3n-digital ê¥Ø3²ê. =,  § an = m2 (5-6) vkê). 3©z [44] ¥, əïÄ ù˜¯K¿y² :  n ´²ê Ú ²Ïfêž, a Ø´²ê. éÙ{ê n, Ü©+Ç ߎ´Ä(E´˜‡úm¯K. C, ² [45] éù˜¯K?1 ïÄ, ?˜ÚÜ©)û ù˜ßŽ, =‰Ñ e¡½n: ½n 5.3. § (5-6) kê), ÙÜ©)ŒL«Ǒ: n n= n12 · (10p (p−1) i+k0 + 3) , p2 71 'uSmarandache¯KïÄ#? √ √ 30 p 3p < n1 < , i = 0, 1, 2, · · · . + 3), 30 3 Ù¥ p | (10 ½n 5.4. é?¿ê k ≥ 1, 3 Smarandache kn-digital ê  {a (n)} ¥3 ‡²ê. Ù¥, § a (n) = m Ü©) ŒL«Ǒ 2 p (p−1) i+k0 k 2 k Ù¥ n12 · (10p (p−1) i+k0 + k) , n= p2 √ √ 10 k p kp < n1 < , i = 0, 1, 2, · · · . p2 | (10p (p−1) i+k0 + k), 10 k k d±þ½n´ѱeíØ: íØ 5.2.1. é?¿ê b, e b 2 | (10k0 + 3), K§ (5-6) )Ǒ n12 · (10k0 + 3) , n= b2 Ù¥ √ √ 30 b 3b < n1 < . 30 3 Ǒ íØ 5.2.2. é?¿ê b, e b n= Ù¥ √ √ 10k b kb < n1 < . 10k k 2 | (10k0 + k), K§ (5-6) ) n12 · (10k0 + k) , b2 Ǒ ¤½ny², ·‚I‡±eA‡Ún: Ún 5.2.1. éƒê p, e p | (10 + 3), K p 2 k0 2 i = 0, 1, 2, · · · . | (10p (p−1) i+k0 + 3), y²: ´ p | (10 +3) ž, (p 6= 2, 5), u´ (10, p ) = 1. d Euler ½n,  10 ≡ 1 (mod p ). 5¿ p | (10 + 3), u´ 10 ≡ −3 (mod p ), Ù¥ i = 0, 1, 2, · · · , l p | (10 + 3), Ù¥ i = 0, 1, 2, · · · . k φ(p2 ) 2 2 p (p−1)i+k0 2 72 p (p−1)i+k0 2 2 k0 1ÊÙ 'u Smarandache ê9Ùk'¯K ùÒ¤ Ún 5.2.1 y². Ún 5.2.2. éƒê p, e p ∤ (10 − 1), K3˜‡ ê p δ  10 ≡ 1 (mod p ). y²: - δ = min{ d : 10 ≡ 1 (mod p), d | ( p − 1 ) }. ÏǑ p ∤ (10 − 1), ¤± p ∤ (10 − 1) 2 pδ p−1 2 d p−1 2 2 δ 1 + 10δ + 102δ + · · · + 10(p−1)δ ≡ p ≡ 0 (mod p), (10δ − 1) (10(p−1)δ + 10(p−2)δ + · · · + 10δ + 1) ≡ 10p δ − 1 ≡ 0 (mod p2 ). XJ3,˜‡ê u  p | (10 − 1) u < p δ, K δ < u < ≡ p δ. w, δ | u. - u = k δ(1 < k < p), ÏǑ 1+10 +10 +· · ·+10 k 6≡ 0 (mod p), ¤± p ∤ (1 + 10 + 10 + · · · + 10 ), p ∤ (10 − 1), u´k p ∤ (10 − 1)(1 + 10 + 10 + · · · + 10 ), = p ∤ (10 − 1), gñ. ùÒ¤ Ún 5.2.2 y². Ún 5.2.3. 3ƒê p Úê k  p | (10 + 3). y²: é?¿ê k, ò k ©ǑXen‡«m: A = {k | 10 + 3 = p p · · · p , Ù¥–3˜‡α ≥ 2, 1 ≤ i ≤ r}, B = {k | 10 + 3 = p p · · · p , Ù¥–3˜‡p , 1 ≤ i ≤ r÷v p ∤ (10 − 1)}, C = {k | 10 + 3 = p p · · · p , é?¿p , 1 ≤ i ≤ r, ÷v p | (10 − 1)}. ·‚©±eA«œ¹5?Ø: œ¹ 1. XJ k ∈ A, K3˜‡ê α ≥ 2 (1 ≤ i ≤ r), u ´ p | (10 + 3). Ún 5.2.3 y. œ¹ 2. XJ k ∈ B, 3 p , p , . . . , p ¥–3˜‡ƒê p,   p ∤ (10 − 1). ´ p | (10 + 3) (p, 10) = 1. dÚn 5.2.2 , 2 u δ δ 2 δ 2δ δ 2δ α1 α2 1 2 k 2 i (k−1)δ (k−1)δ 2 (k−1)δ kδ k0 αr r 1 2 δ 2 2 0 k 2δ i i r p−1 k 1 2 r 2 i i p−1 i 2 i k 1 2 p−1 2 r k p2 ∤ (10δ − 1) Ù¥ i = 0, 1, 2, · · · , k ≡ k (mod δ). é?¿ i = 0, 1, 2, · · · , p − 1, 10 p + 3 H{ p {X. 10δ i+k1 ≡ −3(mod p), 1 δ i+k1 73 'uSmarandache¯KïÄ#? d , e3 i, j  10 p + 3 ≡ 10 p + 3 (mod p), Ù ¥ 0 ≤ i < j < p − 1, K p | 10 (10 − 1), u´ p | (10 − 1). =, 10 ≡ 1 (mod p ), 1 ≤ j − i ≤ p − 1. dÚn 5.2.2 , p δ ´/ X 10 ≡ 1 (mod p ) ¥ê, p δ | δ(j − i), = p | (j − i),  gñ. u´·‚˜‡ i (0 ≤ i < p−1)  10 p + 3 ≡ 0 (mod p), = p | (10 + 3), e k = δ i + k , K δ j+k1 δ i+k1 2 δ(j−i) δ i+k1 δ(j−i) 2 δ(j−i) 2 pδ 2 δ i0 +k1 0 2 0 δ i0 +k1 0 0 1 p2 | (10k0 + 3). œ¹ 3. é?¿ƒê p, 3 p , p , . . . , p ¥, XJ k ∈ C p | (10 − 1), K 10 + 3 ≡ 10 + 3 (mod p ) (j = 0, 1, · · · ). =, p ∤ (10 + 3), j = 0, 1, · · · . nþ, ´ A 6= Ø ½ B 6= Ø. d , k ∈ C 10 + 3 = p p · · · p . é?¿ p (1 ≤ i ≤ r), p | (10 − 1). ù´ØŒU. ~X,  k = 34 ž 49 | (10 + 3), k ∈ A.  k = 1 ž k ∈ B, ùÒ ¤ Ún 5.2.3 y². ½ny²: e¡·‚ò¤½ny². Äk, y²½n 5.3. - n ´ k ?›ê, u´l {a } ½Â, 1 p−1 2 (p−1)j+k 2 2 r k 2 (p−1)j+k k 1 2 r 2 i i pi −1 34 n an = n · (10k+1 + 3), ½ an = n · (10k+2 + 3). (Üə [44] (J, N´ѱe(Ø: XJ 10 +3 ½ 10 +3 ´˜‡ ²Ïfê, K a Ø´˜‡²ê. l , e a ´˜‡ ²ê, K 10 + 3 ½ 10 + 3 7L´²ê. y3, ·‚ÏL² ê 10 + 3 ½ 10 + 3 E§ (5-6) ê). lÚn 5.2.1 ÚÚn 5.2.3 , 3ƒê p Úê k  p | (10 + 3), Ù¥ i = 0, 1, 2, . . . . k+1 k+2 n k+1 k+1 n k+2 k+2 0 2 74 p (p−1) i+k0 1ÊÙ 'u Smarandache ê9Ùk'¯K e 10p (p−1) i+k0 + 3 , p2 √ √ 3 n2 30 p 3p < n1 < ), 3n = 21 ·(10p (p−1) i+k0 +3), 30 3 p n = n12 · = 1 3 n2 < 21 < 1 ( 10 p u´ K 10p (p−1) i+k0 + 3 · (10p (p−1) i+k0 + 3) p2 10p (p−1) i+k0 + 3 2 = n12 · p2 · ( ) . p2 an = n12 · e 10p (p−1) i+k0 + 3 , m = n1 · p Ù¥ √ √ 30 p 3p < n1 < , i = 0, 1, 2, · · · , 30 3 (5-7) 5.3 . u´ (5-7) ´§ (5-6) ), ùÒ¤ ½n y² ^aq {=Œy²½n 5.4. 75 'uSmarandache¯KïÄ#? 18Ù ˜ ¹ Smarandache ¼ê§ 6.1 ¹– Smarandache ¼êÚ Smarandache LCM ¼ê§ 31Ù¥·‚‰Ñ Smarandache LCM ¼ê½Â, =é? ¿ê n, SL(n) ½ÂǑê k,  n | [1, 2, · · · , k], ù p [1, 2, · · · , k] L« 1, 2, · · · , k úê. ØJy,  n I O©)ªǑ n = p p · · · p ž, αk k α1 α2 1 2 SL(n) = max{pα1 1 , pα2 2 , · · · , pαk k }. e¡·‚0 ±eù‡¼ê: ½Â 6.1. – Smarandache ¼ê Z(n) ½ÂǑê k,   n | k(k 2+ 1) , =  m(m + 1) Z(n) = min m : m ∈ N, n | 2  . 'uù‡¼ê5Ÿ, Ǒ,·‚–8Øõ, ®áÚ ØÆö?1ïÄ, ¿¼ ˜ kdŠïĤJ. ~X, Kenichiro Kashihara Ú David Gorski ïÄ ¼ê Z(n) 5Ÿ, ¿ y² ˜ k(Jµ é?¿ƒê p ≥ 3, Z(p) = p − 1; é?¿ƒê p ≥ 3 Ú?¿ k ∈ N, Z(p ) = p − 1; é?¿ k ∈ N, Z(2 ) = 2 − 1; é?¿ê k > 0, XJ n ØUL«Ǒ 2 /ª, o Z(n) < n. oçï [46] ïÄ § k k k k+1 k Z(n) = SL(n), Z(n) + 1 = SL(n) Œ)5, ¿|^9)ې{¼ ùü‡§¤kê). äN/`ǑÒ´y² e¡ü‡(Ø: 76 18Ù ˜ ¹ Smarandache ¼ê§ ½n 6.1. é?¿ê n > 1, § Z(n) = SL(n) ¤á = n = p · m, Ù¥ p Ǒۃê, a ≥ 1, 9 m Ǒ p 2+ 1 ? ¿Œu 1 Ïê. ½n 6.2. é?¿ê n > 1, § a a Z(n) + 1 = SL(n) ¤á = n = p · m, Ù¥ p Ǒۃê, a ≥ 1, 9 m Ǒ p 2− 1 ? ¿Ïê. y²: ±e·‚ò†‰Ñ½ny². Äk·‚y²½n 6.1.  n = 1, w,k Z(n) = SL(n). ²y n = 2, 3, 4, 5 ž, n Ø÷ v§ Z(n) = SL(n), u´b½ n ≥ 6 ÷v§ Z(n) = SL(n), Ø ” n = p p · · · p Ǒ n IO©)ª, p < p < p · · · < p , ¿ - Z(n) = SL(n) = k, d¼ê Z(n) 9 SL(n) ½ÂŒ k ´ ê n ÷ve¡ü‡ت: a a a1 a2 1 2 ar r 1 n | [1, 2, · · · , k], n | 2 3 r k(k + 1) . 2 d ¼ ê SL(n)  5 Ÿ: é ? ¿   ê n, k SL(n) = max {p , p , · · · , p }, ddŒ±íÑ k = p . I.  k ´Ûêž: (1)  a = 1, ´ Z(n) = SL(n) = p, Šâ SL(n) þã5 Ÿ, Œ- n = p · m, m = p p · · · p , Ù¥ a ≥ 0 p ≥ p , i = a1 1 a2 2 ar r a a1 a2 1 2 ai i i ai i 1, 2, 3, · · · , r − 1. džw,k SL(n) = p, qk n | k(k 2+ 1) , Œ, p · m | p(p 2+ 1) , = m | p +2 1 .  m = 1 ž, n = p, Z(p) = p − 1, SL(p) = p w, Z(n) 6= SL(n), ¤± m = 1 Ø÷v§.  m 6= 1 ž, SL(p · m) = p Z(p · m) = p, ù´ÏǑ m Ø p(p − 1) Ø 2 , ÄK† m | p +2 1 gñ. 77 'uSmarandache¯KïÄ#? ¤± Z(m · p) = SL(m · p) = p,  n = p · m, m | (p +2 1) m > 1. (2)  a 6= 1, ·‚k Z(n) = SL(n) = p , ÓnŒ- n = p ·m , m = p · · · p , Ù¥ a ≥ 0 p ≥ p , i = 1, 2, 3, · · · , r − 1. d (1) , a pa11 a22 ai i i a a 1 1 ai i pa + 1 , m1 | 2 m1 = 1 , n = pa , Z(pa ) = pa − 1, SL(pa ) = pa , Z(n) 6= m1 = 1 . SL(n), a m1 6= 1 , SL(p ·m1 ) = pa Z(pa ·m1 ) = pa , m1 a a a p − 1 pa + 1 p (p − 1) , , , (m1 , pa ) = 1, , m1 | m1 | 2 2 2 . (pa + 1) Z(pa · m1 ) = SL(pa · m1 ), n = pa · m1 , m1 | 2 m1 > 1. II. k : Z(n) = SL(n) = k, k = pa : 2×3 , (1) a = 1, Z(n) = SL(n) = 2, n| n | 3, 2 n=1 n = 3, n≥6 . p=2 , . a a (2) a 6= 1, Z(n) = SL(n) = 2 , n = 2 ·m2 , m2 = a1 a2 ai ai a p1 p2 · · · pi , ai ≥ 0 2 ≥ pi , i = 1, 2, 3, · · · , r − 1. : a a a (2 − 1) 2 (2 − 1) , . m| 2a m | 2 2 a m=1 ,n=2 , Z(2a ) = 2a+1 − 1, SL(2a ) = 2a , 2a+1 − 1 = 2a , a = 0, n = 1, n≥6 . p=2 . m 6= 1 , SL(2a · m) = 2a , Z(2a m) = 2a , a a 2 (2 − 1) Z(n) : 2a m | 2m | 2a − 1, . , 2 p = 2a .  ž w, ¤± Ø÷v§  ž ù´ÏǑ Ø Ø ÄK d ´ ù† gñ ¤±   ´óêž d   ·‚k du =  ½ w,ù† gñ ¤± ž § )  ·‚k ÓnŒÙ¥ Xþã =  ž w,k ‡÷ v§7Lk =  ù† gñ ¤ ± § )  ž w, ‡÷v§I‡ d ½Â u´k w,Ø¤á ¤± § ) y3y²½n 6.2. †½n 6.1 y²{ƒq, ùp‰ÑŒV y²L§. w, n = 1, 2 Ø÷v§ Z(n) + 1 = SL(n). u´b½ n ≥ 3 ž ÷v§ Z(n) + 1 = SL(n), ؔ n = p p · · · p Ǒ n IO ©)ª, ¿- Z(n) + 1 = SL(n) = k, d¼ê Z(n) 9 SL(n) ½ÂŒ  k ´ê n ÷ve¡ü‡ت: a1 a2 1 2 n | [1, 2, · · · , k], n | 78 k(k − 1) . 2 ar r 18Ù ˜ ¹ Smarandache ¼ê§ d¼ê SL(n) 5Ÿ: ddŒ±íÑ k = p I.  k ´Ûêž: (1)  a = 1, ·‚k Z(n) + 1 = SL(n), Šâ SL(n) þã5 Ÿ, Œ- n = p · m, m = p p · · · p , Ù¥ a ≥ 0 p ≥ p , i = a a1 a2 1 2 ai i ai i i 1, 2, 3, · · · , r − 1. džw,k SL(n) = p, qk n | k(k 2− 1) , Œ, p · m | p(p 2− 1) , = m | p −2 1 .  m = 1 ž, n = p, Z(p − 1) = p − 1, SL(p) = p, w, Z(n) + 1 = SL(n), ¤± m = 1 ÷v§.  m 6= 1 ž, SL(p · m) = p Z(p · m) = p − 1, ù´ÏǑd pm | p(p − 1)  Z(n) ≤ p − 1, Šâ Z(n) 5Ÿ: Z(n) ≥ max{Z(m) : 2 m | n}, ´ Z(n) ≥ Z(p) = p − 1,  Z(p) = p − 1. dž÷v § Z(n) + 1 = SL(n). (2)  a 6= 1, ·‚k Z(n) + 1 = SL(n) = p , ÓnŒ- n = p ·m , m = p p · · · p , Ù¥ a ≥ 0 p ≥ p , i = 1, 2, 3, · · · , r−1. džw,k SL(p) = p Ú m | p 2− 1 .  m = 1 ž, n = p , Z(p ) = p − 1, SL(p ) = p , w, Z(n)+1 = SL(n), ¤± m = 1 ÷v§.  m 6= 1 ž, SL(p · m ) = p Z(p · m ) = p − 1, ù´Ï Ǒd p m | p (p 2− 1) , Z(n) ≤ p − 1, Šâ Z(n) 5Ÿ: Z(n) ≥ max{Z(m) : m | n}, ´ Z(n) ≥ Z(p ) = p − 1,  Z(p ) = p − 1. d ž÷v§ Z(n) + 1 = SL(n). II.  k ´óêž: d Z(n) + 1 = SL(n) = k, k = p : 1×2 ,n=1 (1)  a = 1, ·‚k Z(n) + 1 = SL(n) = 2, du n | 2 † n ≥ 3, gñ. w,Ø÷v§. ¤± p = 2 ž, § ). (2)  a 6= 1, ·‚k Z(n) + 1 = SL(n) = 2 , ÓnŒ- n = 2 ·m , m = p p · · · p , Ù¥ a ≥ 0 2 ≥ p , i = 1, 2, 3, · · · , r−1. Xþã: 2 m | 2 (2 2− 1) , = m | (2 2− 1) . ¤± p = 2 ž, § ). nܱþA«œ¹, ·‚áǑ¤½n 6.2 y². a a 1 a1 a2 1 2 1 ai i i a 1 a 1 ai i a a a a a a 1 a 1 a a a a 1 a a 1 a 1 a a a a a a a 2 2 a a1 a2 1 2 a a ai i i a a ai i 'u– Smarandache ¼ê Z(n) Ú Smarandache LCM ¼ê SL(n), 79 'uSmarandache¯KïÄ#? Ç! [47] ǑéÙ?1 ïÄ, ¿‰Ñ § Z(n) + SL(n) = n ¤kê), =‰Ñ e¡½n: ½n 6.3. é?¿ê n, § Z(n) + SL(n) = n ¤kê)ŒL«Ǒ: n = 2 p , Ù¥ p > 2 ´ƒê, k Ú α ´÷v ±e^‡ê: 1.  2 > p ž, p | (2 − 1). 2.  2 < p ž, 2 | (p − 1), 2 ∤ (p − 1). y²: e¡·‚ò^{¿(ÜÓ{gŽ¤½ny². w, n = 6 ´§ Z(n) + SL(n) = n ˜‡). y3·‚b ½ n = 2 · s, Ù¥ s ǑÛê, e¡©A«œ¹5?Ø: (a). e n ǑÛê, K k = 0, n = s. (1) - s = 1, K Z(1) = 1, SL(1) = 1. u´ Z(1) + SL(1) = 2 6= 1. (2) - s = p, p ´Ûƒê, K SL(p) = p, Z(p) = p − 1. u´ k α k α α k k α k α k+1 α k Z(p) + SL(p) = 2p − 1 6= p. ´Ûƒê Ǒê u´ l k Ù¥ ê ´ Œƒê˜ = (3) s = pα , p ,α , SL (pα ) = pα , Z(pα ) = pα − 1. Z(pα ) + SL(pα ) = 2pα − 1 6= pα . s = pα · pα1 1 pα2 2 · · · pαr r , p, p1 , p2 , · · · , pr ,α (4) α ,p s . , ´Ûƒê Ǒ pα = max {pα , pα1 1 , pα2 2 , · · · , pαr r } . - p p · · · p = t, K s = p · t. u´ SL(n) = p . e Z(n) = n − SL(n) = p (t − 1), d Z(n) ½Â, α1 α2 1 2 αr r α α α pα · t | pα (t − 1)[pα (t − 1) + 1] . 2  t | (p − 1). dž, e m = p − 1, ÓŒ n = p · t  Ø m(m2+ 1) . 5¿ p − 1 < p (t − 1). Ï 3dœ¹e§ ). α α α 80 α α 18Ù ˜ ¹ Smarandache ¼ê§ d (1)-(4) Œ§ Ûê). (b). e n Ǒóê, K k 6= 0. (1) - s = 1, K n = 2 , u´ Z(2 ) = 2 − 1, SL(2 ) = 2 . l k k k+1 k k Z(2k ) + SL(2k ) = 3 · 2k − 1 6= 2k . - s = p, K n = 2 · p, p Ǒۃê, k Ǒê. dž, e 2 > p, K SL(n) = 2 , e÷v Z(n) + SL(n) = n, K k (2) k k Z(n) = m = n − SL(n) = 2k p − 2k = 2k (p − 1). d Z(n) ½ÂŒ, 2k · p | 2k (p − 1)(2k (p − 1) + 1) . 2 u´Œ p | (2 − 1). y3·‚5y² m = 2 (p − 1) ´÷v Z(n) ½ÂŠ. d Z(n) 5Ÿ ,  Z(n) ≥ 2 − 1, Z(n) ŒU p−1 Š0u 2 − 1 Ú 2 · 2 ƒm, kXeü«: k k k+1 k+1 k+1 p−1 − 1. 2 p−1 }. 2k+1 · s1 − 1, s1 ∈ {1, 2, · · · , 2 A. 2k+1 − 1, 2k+1 · 2 − 1, · · · , 2k+1 · Pù|êǑ Œ 1 ≤ 2s 1 B. 2 k+1 k+1 , 2 Pù|êǑ 2k+1 · s1 − 1 ≡ 2s1 − 1 (mod p). − 1 ≤ p − 2. k+1 u´ p ∤ (2 p−1 k+1 · s1 − 1). − 1). 2 p−1 − 1}. 2k+1 · s2 , s2 ∈ {1, 2, · · · , 2 · 2, · · · , 2 ·( 2k+1 · s2 ≡ 2s2 (mod p). Œ 2 ≤ 2s ≤ p − 3. u´ p ∤ 2 · s . e 2 < p, K SL(n) = p, e÷v Z(n) + SL(n) = n, K 2 k+1 2 k Œ Z(n) = m = n − SL(n) = p(2k − 1). n = 2k · p | p(2k − 1)(p(2k − 1) + 1) . 2 81 'uSmarandache¯KïÄ#? Ï , 2 | [(2 − 1)p + 1]. =, 2 | (p − 1), 2 ∤ (p − 1). ey m = p(2 − 1) ´÷v Z(n) ½ÂŠ. d Z(n) 5Ÿ, Z(n) ≥ p − 1, Z(n) 3 p − 1 Ú p(2 − 1) ƒmŒUŠǑ: C. p − 1, p · 2 − 1, · · · , p · (2 − 1) − 1. Pù|êǑ p · s − 1, s ∈ {1, 2, · · · , 2 − 1}. k+1 k k k+1 k k k 1 k 1 p · s1 − 1 ≡ s1 − 1 (mod 2k ). Œ 0 ≤ s − 1 ≤ 2 − 2.  s − 1 = 0 ž, s | (p − 1). l gñ.  2 ∤ (p · s − 1). D. p, p · 2, · · · , p · (2 − 2). Pù|êǑ p · s , s ∈ {1, 2, · · · , 2 − 2}. k 1 2 k+1 1 1 k = 1, u´ m = p − 1, 1 k 2 k 2 p · s2 ≡ s2 (mod 2k ). Œ 1 ≤ s ≤ 2 − 2. u´ 2 ∤ p · s . (3) - s = p , n = 2 · p , p Ǒۃê, k !α Ǒê. dž, e 2 > p , K SL(n) = 2 .  Z(n) + SL(n) = n ž, Œ  Z(n) = m = n − SL(n) = 2 (p − 1). d Z(n) ½Â, k 2 α k k k 2 α α k k 2k · pα | α 2k (pα − 1)(2k (pα − 1) + 1) . 2 Ïd p | (2 − 1). ey m = 2 (p − 1) ´÷v Z(n) ½ÂŠ . d Z(n) 5Ÿ, p −1 k Z(n) ≥ 2 − 1, Z(n) 3 2 − 1 Ú 2 · 2 ƒmŒUŠk: α k k α k+1 k+1 k+1 α pα − 1 − 1. 2α p −1 }. 2k+1 · s1 − 1, s1 ∈ {1, 2, · · · , 2 A. 2k+1 − 1, 2k+1 · 2 − 1, · · · , 2k+1 · Pù|êǑ Œ 1 ≤ 2s 1 2k+1 · s1 − 1 ≡ 2s1 − 1 (mod pα ). − 1 ≤ pα − 2. u´ p ∤ (2k+1 · s1 − 1). p −1 − 1). 2 α p −1 − 1}. 2k+1 · s2 , s2 ∈ {1, 2, · · · , 2 B. 2k+1, 2k+1 · 2, · · · , 2k+1 · ( Pù|êǑ α α 2k+1 · s2 ≡ 2s2 (mod pα ). 82 18Ù ˜ ¹ Smarandache ¼ê§ Œ 2 ≤ 2s ≤ p − 3. u´ p ∤ 2 · s . e 2 < p , K SL(n) =
p , d Z(n) + SL(n) = n ŒíÑ Z(n) = m = n − SL(n) = p 2 − 1 . Ï , α 2 k α k+1 2 α k α

pα 2k − 1 (pα 2k − 1 + 1) n=2 ·p | . 2 k α dd´íÑ 2 | (p − 1) 2 ∤ (p − 1). ey m = p (2 − 1) ´÷v Z(n) ½ÂŠ. d Z(n) ½Â, k Z(n) ≥ p − 1, Z(n) 3 p − 1 Ú p (2 − 1) ƒmŒUŠǑ: C. p − 1, p · 2 − 1, · · · , p · (2 − 1) − 1. Pù|êǑ p · s − 1, s ∈ {1, 2, · · · , 2 − 1}. k α α k+1 α k α α α α α α α 1 k k k 1 pα · s1 − 1 ≡ s1 − 1 (mod 2k ). Œ 0 ≤ s − 1 ≤ 2 − 2.  s − 1 = 0 ž, s = 1, m = p | (p − 1). dd·‚gñ. Ï 2 ∤ (p · s − 1). D. p , p · 2, · · · , p · (2 − 2). Pù|êǑ p · s , s ∈ {1, 2, · · · , 2 − 2}. k 1 2 k+1 1 α α k α α α 2 α 1 α − 1, 1 k k 2 pα · s2 ≡ s2 (mod 2k ). Œ 1 ≤ s ≤ 2 − 2. u´ 2 ∤ p · s . (4) - n = 2 · s, Ù¥ s = p · p p · · · p , p, p , p , · · · , p ´ ۃê, α Ǒê, p ´ s Œƒê˜. =, 2 k k k α α 2 α1 α2 1 2 αr r 1 2 r α pα = max {pα , pα1 1 , pα2 2 , · · · , pαr r } . 3ù«œ¹e, ·‚y²e n k–n‡ØӃÏfž, § Z(n)+ SL(n) = n ). - a = 2 · p p · · · p , K n = 2a · p , α ≥ 1, (2a, p ) = 1, p Ǒ ƒê p ≥ 3. e¡·‚©ü«œ¹5?Ø: e 2 > p , K SL(n) = 2 . = Z(n) = n − 2 ž§kê). l (2a, p ) = 1 ·‚Ó{§ k−1 k α α1 α2 1 2 αr r α k α k α 4ax ≡ 1 (mod pα ) 83 'uSmarandache¯KïÄ#? kê), u´Ó{§ 16a2 x2 ≡ 1 (mod pα ) kê). )Ǒ y,  p1 ≤−y1≤ p − 1, K p − y ½Ǒ§). l Œ 1 ≤ y ≤ 2 . l 16a y ≡ 1 (mod p ) Œ p (4ay − 1) ½ p | (4ay + 1). A. e p | (4ay − 1), K α α α 2 2 α α α | α n = 2a · pα | Œ 4ay(4ay − 1) . 2 4a(pα − 1) Z(n) = m ≤ 4ay − 1 ≤ − 1 = n − 2a − 1. 2 B. ep α | (4ay + 1), K n = 2a · pα | Œ 4ay(4ay + 1) . 2 4a(pα − 1) = n − 2a. Z(n) = m ≤ 4ay ≤ 2 w, 2 < a, Z(n) = n − 2 , Z(n) > n − a. dž§ ). e 2 < p , K SL(n) = p . = Z(n) = n − p = p (2a − 1) ž §kê). l (2a, p ) = 1 ŒÓ{§ k k k α α α α α kê), u´Ó{§ pα x ≡ 1 (mod 2a) p2α x2 ≡ 1 (mod 2a) kê). )Ǒ2ay, −1 1 ≤ y ≤ 2a − 1, K 2a − y ǑǑ§ê ). Œ 1 ≤ y ≤ 2 . d p x ≡ 1 (mod 2a) Œ 2a | (p y − 1) ½ 2a | (p y + 1). 2α 2 84 α α 18Ù ˜ ¹ Smarandache ¼ê§ C. e 2a | (p y − 1), K α y Ǒóê. Œ D. y n = 2a · pα | pα y(pα y − 1) . 2 Z(n) = m ≤ pα y − 1 ≤ pα · e 2a | (p y + 1), K 2a − 1 − 1. 2 α Ǒóê. Œ n = 2a · pα | pα y(pα y + 1) . 2 Z(n) = m ≤ pα y ≤ pα · 2a − 1 . 2 u´ Z(n) = n − p Ø´÷v Z(n) ½ÂŠ. dž§  ê). ùÒ¤ ½ny². α 6.2 ˜‡¹ Smarandache ¼ê†– Smarandache ¼ê§ þ˜!ïÄ ü‡¹ Smarandache LCM ¼ê†– Smarandache ¼ê§, ùÜ©·‚UY&?§Œ)5¯K. Äu©z [48]  ÄgŽ, ·‚̇0 o ڃ‘|ïĤJ, =|^Ú|Ü {ïļꐧ Z(n) + S(n) = kn (6-1) Œ)5, Ù¥ k Ǒ?¿ê. äN`Ò´y² e¡½n: ½n 6.4.  k = 1 ž, n = 6, 12 ´§ (6-1) =kü‡AÏ ê); džÙ§ê n ÷v§ (6-1)  p −= n = p·u ½ 1 ö n = p · 2 · u, Ù¥ p ≥ 7 Ǒƒê, 2 | p − 1, u ´ 2 ?¿˜‡Œ u 1 ÛêÏf. α α α 85 'uSmarandache¯KïÄ#? ½n 6.5.  k = 2 ž, n = 1 ´§ (6-1) ˜‡AÏ); Ù§ ê n ÷v§ (6-1)  = n = p · u, Ù¥ p ≥ 5 Ǒƒê, u ´ p −2 1 ?¿˜‡óêÏf. 5¿, Z(n) ≤ 2n − 1 9 S(n) ≤ n, ¤± k > 2 ž, § (6-1) v kê). l½n 6.4 éN´éŽ Fermat ƒê, =/X F = 2 + 1 ƒê, Ù¥ n ≥ 1 Ǒê. ~X, F = 5, F = 17, F = 257, . d½ n 6.4 ØJíÑe¡íØ: íØ 6.2.1.  k = 1 ž, XJ n ¹k Fermat ƒÏf, K n ،U ÷v§ (6-1). ½ny²: ·‚|^9|ܐ{5¤½ny². Äky ²½n 6.4. ùž k = 1. 5¿ Z(1)+S(1) = 2 6= 1, Z(2)+S(2) = 5 6= 3, Z(3) + S(3) = 5 6= 3, Z(4) + S(4) = 11 6= 4, Z(5) + S(5) = 9 6= 5, Z(6) + S(6) = 6, ¤± n = 1, 2, · · · , 5 Ø÷v§ (6-1), n = 6 ÷v § (6-1), u´Ù§ n ÷v§ (6-1) ž˜½k n ≥ 7,  n = p p · · · p Ǒ n IO©)ª, džd Smarandache ¼ê5Ÿ n 1 α1 α2 1 2 2 2n 3 αk k S(n) = max {S(pαi i )} = S(pα ) = u · p, 1≤i≤k Ù¥ p Ǒ,˜ p , α Ǒ,˜ α , u ≤ α. y35¿ p | n 9 S(n) = u · p, ¤±Œ n = p § (6-1) žk i i α · n1 .  n ÷v Z(n) + u · p = pα · n1 . (6-2) Äky²3 (6-2) ª¥ α = 1. ÄKb½ α ≥ 2, u´d (6-2) ªáǑ íÑ p | Z(n) = m. d Z(n) = m ½Â n = p · n Ø m(m2+ 1) , (m, m + 1) = 1, ¤± p | m. l d (6-2) ªíÑ p | S(n) = u · p, = p | u, l p ≤ u. ´,˜¡, 5¿ S(n) = S(p ) = u · p, d Smarandache ¼ê S(n) 5Ÿ u ≤ α, ¤± p ≤ u ≤ α. dªé ۃê p w,ؤá. XJ p = 2, K α ≥ 3 ž, p ≤ u ≤ α Ǒؤ á. u´k˜«ŒU: u = α = 2. 5¿ n ≥ 5 ±9 S(n) = 4, ¤±d žk˜«ŒU: n = 12, n = 12 ´§ (6-1) ˜‡). ¤±XJÙ §ê n ÷v§ (6-1), K (6-2) ª¥7k S(n) = p, α = u = 1. 3 ù«œ¹e, - Z(n) = m = p · v, K (6-2) ª¤Ǒ α 1 α α−1 α α−1 α α−1 α−1 v + 1 = n1 , 86 18Ù ˜ ¹ Smarandache ¼ê§ ½ö n = v + 1, = n = p · (v + 1), Z(n) = p · v. 2d Z(n) ½Â  n = p · (v + 1) Ø 1 Z(n)(Z(n) + 1) pv · (pv + 1) = , 2 2 ½ö (v + 1) Ø Z(n)(Z(n) + 1) v · (pv + 1) = . 2 2 5¿ (v + 1, v) = 1, ¤± v ǑóêždþªáǑíÑ v+1 | p−1 p−1 . w,é pv + p − p + 1, = v + 1 | p − 1 ½ö v + 1 | 2 ?¿Œu 1 ÛêÏf r, n = p · r ´§ (6-1)2 ). ÏǑdž k Z(p · r) = p · (r − 1).  v ǑÛêž, d (v + 1) Ø Z(n)(Z(n) + 1) v · (pv + 1) = , 2 2  ddŒíÑ v+1| (pv + 1) (p − 1)(v + 1) + v − p + 2 = . 2 2 p − 1 = (2k + 1) · (v + 1). u´ p − 1 = 2 · h, Ù¥ h ǑÛê, K v 2+ 1 Ǒu h ÛêÏf. N ´yé?¿Ûê r | h r < h, n = p · 2 · r Ǒ§ (6-1) ). ÏǑ džk β β β Z(p · 2β · r) = p · (2β · r − 1). ¯¢þ, 5¿ r | h, ÄkN´íÑ p · 2 β ·r Ø p · (2β · r − 1) · (p · (2β · r − 1) + 1) . 2 Ùg m < p · (2 d Z(n) ½Â β · r − 1) ž, ،Uk p · 2 β ·r Ø m(m2+ 1) . u´ Z(p · 2β · r) = p · (2β · r − 1). 87 'uSmarandache¯KïÄ#? u´y² ½n 6.4. y3y²½n 6.5. dž5¿ k = 2, ¤± n = 1 ž, k Z(1) + S(1) = 2, = n = 1 ´§ (6-1) ˜‡). XJ§ (6-1) „kÙ§ ê) n > 2, Kd½n 6.4 y²{ØJíÑ n = p · u, Ù¥ p ≥ 5 Ǒƒê, S(u) < p. “\§ (6-1) Œ Z(p · u) + S(p · u) = 2p · u. ddªáǑíÑ p Ø Z(p·u).  Z(p·u) = p·v, K v = 2u−1. d Z(n) − 1) + 1) p−1 ½Â p · u Ø p(2u − 1)(p(2u , l u Ø . d , 2 2  u Ǒ p −2 1 ?˜Œu 1 ÛêÏêž, Z(p · u) = p · (u − 1), ¤±d ž n = p · u Ø´§ (6-1) ê);  u Ǒ p −2 1 ?˜óêÏê žk Z(p · u) = p · (2u − 1), dž Z(p · u) + S(p · u) = 2p · u. u´¤ ½n 6.5 y². d½n 6.4 ØJíÑ©¥íØ. ¯¢þ½n 6.4 ¥ƒê،U ´ Fermat ƒê, ÏǑ p Ǒ Fermat ƒêž, p − 1 vkŒu 1 ÛêÏ f. 6.3 'u Smarandache ¼êü‡ßŽ ©z [49] Ú? Smarandache p‡¼ê S (n): ½Â 6.2. S (n) ½ÂǑ÷v y | n! 1 ≤ y ≤ m Œê m, c = c Sc (n) = max{m : y | n!, 1 ≤ y ≤ m, m + 1 ∤ n!}. ~X, S (n)  A‡ŠǑ: S (1) = 1, S (2) = 2, S (3) = 3, S (4) = c c c c c 4, Sc (5) = 6, Sc (6) = 6, Sc (7) = 10, Sc (8) = 10, Sc (9) = 10, Sc (10) = 10, Sc (11) = 12, Sc (12) = 12, Sc (13) = 16, Sc (14) = 16, Sc (15) = 16, · · · . 88 18Ù ˜ ¹ Smarandache ¼ê§ ©z [49] ïÄ S (n) 5Ÿ, ¿y² ±e(Ø: e S (n) = n 6= 3, K x + 1 ´Œu n ƒê. ©z [50] Ú? – Smarandache éó¼ê Z (n): ½Â 6.3. Z (n) ½ÂǑ÷v P k Ø n Œê m, = c x, c ∗ m ∗ k=1  m(m + 1) |n . Z (n) = max m : 2  ∗ ©z [51] ïÄ Z (n) 5Ÿ,  ˜ ­‡(J. ©z [52] ¥ïÄ ùn‡¼êƒm'X§ Z(n) + Z (n) = n † S (n) = Z (n) + n,  ˜ ­‡(J, ¿JÑ ˜ „™)ûߎ: ߎ 1. § Z(n) + Z (n) = n kk‡óê), ǑN=k˜‡ó ê)Ǒ n = 6. ߎ 2. § S (n) = Z (n) + n )Ǒ p , Ù¥ p Ǒƒê, 2 ∤ α, p + 2 ǑǑƒê.  ïÄ ±þ¯K,  e¡ü‡½n: ½n 6.6.  n Ǒóêž, § Z(n) + Z (n) = n )k n = 6. ½n 6.7. § S (n) = Z (n) + n )Ǒ p , Ù¥ p Ǒƒê, 2 ∤ α, p + 2 ǑǑƒê, ±9÷v^‡ a(2a − 1) ∤ n (a > 1), n + 2 Ǒƒê, n Ǒ ê. 3y²½nƒ , ·‚k‰Ñe¡: Ún 6.3.1. e S (n) = x ∈ Z, n 6= 3, K x + 1 ǑŒu n  ƒê. y²: „©z [49]. ddŒ„, S (n) Ø 3 n = 1, n = 3 ǑÛê , 3Ù{œ¹eŠ Ñ´óê. ∗ ∗ c ∗ ∗ ∗ c α α ∗ c ∗ α α c c 89 'uSmarandache¯KïÄ#? Ún 6.3.2. ( Z ∗ (pα ) = 2, p 6= 3, 1, p = 3. y²: „©z [51]. Ún 6.3.3. e n ≡ 0 (mod a(2a − 1)), Kk Z (n) ≥ 2a > 1. y²: „©z [51]. Ún 6.3.4. √ ∗ 8n + 1 − 1 . 2 Z ∗ (n) ≤ y²: „©z [51]. Ún 6.3.5.  n = p Ǒ n IO©)ªž, k α0 α1 α2 0 p1 p2 Z(n) ≤ n − · · · pαk k (p0 = 2, pi ≥ 3, k ≥ 1, αi ≥ 1) n min{pα0 0 , pα1 1 , pα2 2 , · · · , pαk k } . y²:  n = p p p · · · p (p = 2, p ≥ 3, k ≥ 1, α ≥ 1) ǑÙ IO©)ªž, ©ü«œ¹5y²: (i)  n = 2kp , α ≥ 1, (2k, p ) = 1, p ≥ 3 Ǒƒê, dÓ{§ 4kx ≡ 1 (mod p ) k), ŒÓ{§ 16k x ≡ 1 (mod p ) k), Ù)ؔ Ǒ y, KŒp 1−≤1 y ≤ p − 1, q p − y ½Ǒ ¡Ó{§), KŒ  1 ≤ y ≤ 2 . d 16k x ≡ 1 (mod p ), K p | (4ky − 1)(4ky + 1), (4ky − 1, 4ky + 1) = 1, u´ p | 4ky − 1 ½ p | 4ky + 1. e p | 4ky − 1, K n = 2kp | 4ky(4ky2 − 1) , l αk k α0 α1 α2 0 1 2 α 0 i i α α 2 2 α α α α 2 2 α α α α α α 1 4k(pα − 1) − 1 ≤ n − 2k − 1 ≤ (1 − α )n 2 p n ≤ n− . min{pα0 0 , pα1 1 , pα2 2 , · · · , pαk k } Z(n) = m ≤ 4ky − 1 ≤ ep α | 4ky + 1, K n = 2kp Z(n) = m ≤ 4ky ≤ 90 α | 4ky(4ky + 1) , 2 l Ǒk 1 4k(pα − 1) ≤ n − 2k = (1 − α )n 2 p 18Ù ˜ ¹ Smarandache ¼ê§ ≤ n− n min{pα0 0 , pα1 1 , pα2 2 , · · ·  † ǑÓ{§ =Œ ÄK ÷vÓ{§ §¥7k˜‡÷v ½ e , pαk k } . KÓ{§ þ7k) ǑÛê  ) e K K ü‡Ó{ ) K K n = 2α (2k + 1), (α ≥ 1, k ≥ 1), (2k + 1)x ≡ (ii) 1 (mod 2α+1 ) (2k + 1)x ≡ −1 (mod 2α+1 ) , , α α+1 α (2k + 1)x ≡ 1 (mod 2 ) , 1 ≤ a ≤ 2 − 1, a α α+1 α+1 α+1 α α , 2 +1 ≤ a ≤ 2 − 1, 2 −a ≤ 2 − 2 − 1 = 2 − 1, α+1 α+1 2 −a (2k + 1)x ≡ −1 (mod 2 ), 1 ≤ a ≤ 2α − 1 a, 2α+1 | (2k + 1)a + 1 2α+1 | (2k + 1)a − 1, 2α+1 | (2k + 1)a + 1, 2α+1(2k + 1) | [(2k + 1)a + 1](2k + 1)a, l 2 1 Z(n) ≤ a(2k + 1) ≤ (2α − 1)(2k + 1) ≤ (1 − α )n 2 n ≤ n− . min{pα0 0 , pα1 1 , pα2 2 , · · · , pαk k } α+1 | (2k + 1)a − 1 Z(n) ≤ (1 − ž, ÓnǑk 1 n )n ≤ n − . α α 0 1 2α min{p0 , p1 , pα2 2 , · · · , pαk k } nÜ (i), (ii), ·‚k,  n = p 1, α ≥ 1) ǑÙIO©)ªž, K α0 α1 α2 0 p1 p2 · · · pαk k (p0 = 2, pi ≥ 3, k ≥ i Z(n) ≤ n − n min{pα0 0 , pα1 1 , pα2 2 , · · · , pαk k } . e¡·‚‰Ñ½ny². ½n 6.6 y²: ·‚©ü«œ¹5y². (1)  n –kn‡ØÓƒÏfž, = n = p p p · · · p 2, k ≥ 1, α ≥ 1) ´ Ù I O © ) ª, Ǒ Ö   B, min{p , p , p , · · · , p }, K α0 α1 α2 0 1 2 i α0 0 α1 1 n i p2α i α2 2 αk k α + αk k (p0 = = pαi i α i−1 i+1 pα0 0 pα1 1 pα2 2 · · · pi−1 pi+1 · · · pαk k 1 1 = + αi > 2. αi αi pi pi pi 91 'uSmarandache¯KïÄ#? l †Ún 4n2 4n , 2αi + αi + 1 > 8n + 1, pi pi 6.3.5, k ? n > pαi i n Z(n) + Z (n) ≤ n − αi + pi ∗ √ 8n + 1 − 1 , 2 u´dÚn 6.3.4 √ 8n + 1 − 1 < n. 2 Ǒƒê ©ü«œ¹5y²  K e ´Œu Ûê K † þ´ êg˜ † † pƒgñ l Ǒóê K Œ q u´ e Ǒ§ ) K ? Kk gñ dž Ø´§) e K d k u´ e Ǒ ) K ? § Ï Kk ½  † gñ K l u´ q ùžq©ü«œ¹ e k K k l K = Ǒ§) e K l K ). . (2) n = 2α q β (α ≥ 1, β ≥ 0, q ≥ 3 ∗ α β (i) Z (n) = 2a, a(2a + 1) | 2 q . a 1 , a 2a + 1 q , a 2a + 1 , a , α γ β γ+1 β a|2 , a = 2 (0 ≤ γ ≤ α), (2a + 1) | q , (2 + 1) | q . ∗ α β α β n Z(n) + Z (n) = n , Z(2 q ) = 2 q − 2γ+1 , 2α+1 q β | (2α q β − 2γ+1 )(2α q β − 2γ+1 + 1), q β | 2γ+1 − 1 . n . (ii) Z ∗ (n) = 2a − 1, a(2a − 1) | 2α q β . (a, 2a − 1) = 1, γ β γ+1 a = 2 (0 ≤ γ ≤ α), (2a − 1) | q , (2 − 1) | q β . n ∗ α β α β γ+1 α+1 β Z(n) + Z (n) = n , Z(2 q ) = 2 q − 2 + 1, 2 q | α β γ+1 α β γ+1 α β γ+1 α β γ+1 (2 q −2 +1)(2 q −2 +2). (2 q −2 +1, 2 q −2 +2) = 1, q β | 2γ+1 −1 q β | 2γ+1 −2. q β | 2γ+1 −2 (2γ+1 −1) | q β , β γ+1 β γ+1 α β α β q |2 − 1. q =2 − 1 = 2a − 1, Z(2 q ) = (2 − 1)q . α+1 α β γ+1 2 | (2 q − 2 + 2), . α+1 α β α γ = α, 2 | (2 q +2), 2 | 2, α = 1, a = 2, q β = 3, n = 6. Z(6) + Z ∗ (6) = 3 + 3 = 6, n=6 . γ α−1 β α 0 ≤ γ ≤ α − 1, a = 2 ≤ 2 , q = 2a − 1 ≤ 2 − 1, α β α β ∗ α β Z(2 q ) ≤ 2 (q − 1), Z (2 q ) = q β , Z(2α q β ) + Z ∗ (2α q β ) ≤ 2α (q β − 1) + q β ≤ 2α (q β − 1) + 2α − 1 = n − 1, dž n Ø´§). ½n 6.7 y²: ·‚©Ê«œ¹5y². (1) n = 1 ž, Z (1) = 1, S (1) = 1, K 1 ØǑÙ). (2) n = 3 (α ≥ 1), dÚn 6.3.2, Z (3 ) = 2, e n = 3 ´§ ), K S (3 ) = 2 + 3 , ÏǑ 3 | 3 + 2 + 1, l 3 + 2 + 1 ،UǑ ƒê †Ún 6.3.1 ƒgñ,  n = 3 Ø´§). (3) n = p (α ≥ 1, p ≥ 5Ǒƒê), dÚn 6.3.2, Z (p ) = 1, e n = p ´§), K S (p ) = 1 + p , Ï p ≥ 5 ž, 3 | p + 2, dÚ ∗ c α c ∗ α α α α α α α α α 92 ∗ c α α α 2β 18Ù ˜ ¹ Smarandache ¼ê§ n 6.3.1, α ØUǑóê,  p + 2 (2 ∤ α) Ǒƒêž, n = p (α ≥ 1, p ≥ 5Ǒƒê) ÷v§. m(m + 1) | 2 , Ï (m, m + 1) = 1, K m = 1, (4) n = 2 (α ≥ 1), e 2  Z (2 ) = 1, e n = 2 ´§), K S (2 ) = 1 + 2 , Ï 2 | (2 + 1 + 1), †Ún 6.3.1 gñ,  n = 2 (α ≥ 1) Ø´§). (5) n = p p · · · p (k ≥ 2, α ≥ 1) ǑÙIO©)ª. ·‚q©ü «œ¹5y². (i) 2 ∤ n, K 2 ∤ p ž, l 2 | S (n), e n ‡÷v§, K7 L 2 ∤ Z (n). yÄ Z (n), e3ê a (a > 1),  a(2a − 1) | n, K Z (n) ≥ 2a− 1, e3ê a (a > 1),  a(2a+1) | n, K Z (n) ≥ 2a, l α α α α ∗ α α c α α α α αk k α1 α2 1 2 i αi i ∗ c ∗ ∗ ∗ Z ∗ (n) = max {max {2k : k(2k + 1) | n} , max {2k − 1 : k(2k − 1) | n}} . 2©n«œ¹5?Ø: 1˜, e Z (n) = 2a − 1 > 1, K a(2a − 1) | n, k a | n, a | [n + (2a − 1) + 1]. e n ‡÷v§, K S (n) = 2a − 1 + n. S (n) + 1 ØǑƒê, †Ún 6.3.1 ƒgñ. 1, e Z (n) = 2a > 1, K a(2a + 1) | n, k a | n, (2a + 1) | n. e n ‡÷v§, K S (n) = 2a + n. S (n) + 1 ØǑƒê, †Ún 6.3.1 gñ.  , e Z (n) = 1, d a > 1, K a(2a − 1) ∤ n, l e n + 2 Ø´ ƒê, dÚn 6.3.1, ù n Ø´§). e n + 2 Ǒƒê, dÚ n 6.3.1, ù n Ǒ§). = a(2a − 1) ∤ n, n + 2 Ǒƒêž ê n Ǒ§). Z (n) ≥ 2, (ii) 2 | n, e n ÷v§, K7L Z (n) Ǒóê, m(m + 1) Z (n) = m ≥ 2, | n, K (m+1) | n, ? (m+1) | (n+m+1), 2 ù S (n) = n + m + 1 Ø´ƒê, †Ún 6.3.1 gñ. ùÒ¤ ½ny². ∗ c c ∗ c c ∗ ∗ ∗ ∗ c 6.4 ˜‡¹¼ê S (n) § k !·‚UY0 ¹ Smarandache ¼ê§Œ)5¯K. 93 'uSmarandache¯KïÄ#? ½Â 6.4. éu‰½ê n, k k ≥ 2, Ͷ Smarandache Ceil ¼ê S (n) ½ÂǑê x  n|x , = k k ½Â 6.5.  x |n, = Sk (n) = min{x : x ∈ N, n|xk }. Sk (n) k éó¼ê S (n) ½ÂǑŒê x  k Sk (n) = max{x : x ∈ N, xk |n}. ~X,  k = 2 ž, S (n)  A‡Š´ S (1) = 1, S (2) = 2, S (3) = 3, S (4) = 2, S (5) = 5, S (6) = 6, S (7) = 7, S (8) = 4, S (9) = 3, · · · . S (n)  A‡Š´ S (1) = 1, S (2) = 1, S (3) = 1, S (4) = 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2, S2 (5) = 1, S2 (6) = 1, S2 (7) = 1, S2 (8) = 2, S2 (9) = 3, · · · . 'u S (n) Ú S (n) ˜ 5Ÿ, NõÆöǑ?1 ?Ø, ¿‰ Ñ ˜ k(Ø, k'ù SNë©z [53-55]. ~X, 3©z [53] ¥, [,y² éu÷v (a, b) = 1 ü‡ê a, b, k k k Sk (ab) = max{m : m ∈ N, mk |a}·max{m : m ∈ N, mk |b} = Sk (a)·Sk (b), Ú α Sk (pα ) = p⌊ k ⌋ . Ù¥ ⌊x⌋ ½ÂǑuu x Œê. éu?¿ê n, X J n = p p · · · p L« n IO©)ª, u´ α1 α2 1 2 αr r ⌊ α1 ⌋ ⌊ α2 ⌋ ⌊ αkr ⌋ Sk (pα1 1 pα2 2 · · · pαr r ) = p1 k p2 k · · · pr = Sk (pα1 1 )Sk (pα2 2 ) · · · Sk (pαr r ). lù 5Ÿ·‚Œ± S (n) ´˜‡Œ¼ê, Ӟ·‚Ú?ü ‡¼ê ω(n) Ú Ω(n), e n = p p · · · p , ·‚½Â ω(n) Ǒ n ¤k ØӃÏf‡ê, ؝)ƒÏf­ê, = ω(n) = ω(p p · · · p ) = r. Ω(n) ½ÂǑ n ¤kƒÏf‡êÚ, = Ω(n) = Ω(p p · · · p ) = k α1 α2 1 2 αr r α1 1 α1 1 α2 2 α2 2 αr r αr r α1 + α2 + · · · + αr . § 3˜Ÿ®ÜHŒÆƹ^o¡òØ©¥, ¨ïÄ ¼ê X d|n 94 Sk (d) = ω(n)Ω(n). (6-3) 18Ù ˜ ¹ Smarandache ¼ê§ Œ)5, ¿‰Ñ T§¤kê). äN/`Ò´y² e¡ ½n: ½n 6.8. § (6-3) k ¡õ‡ê), ¿ z‡)áue~œ ¹ƒ˜: 1). n = p p ½ö p p , Ù¥ 1 ≤ α, β ≤ k − 1; 2). n = p p p ½ö n = p p p ½ö n = p p p ; 3). n = p p p p . Ù¥ p < p < p < p Ǒۃê. y²: e¡·‚|^9|ܐ{5¤½ny² . ÄkÏ X Ǒ S (n) ´˜‡Œ¼ê, ¤±dŒ¼ê5Ÿ S (d) Ǒ´Œ . ¼ê, y3Œ±©XeA«œ¹œ¹5y²·‚(Ø X (i).  n = 1 ž, é¤k k ≥ 2, S (d) = S (1) = 1, ω(n)Ω(n) = 0, ª (6-3) ؤá, Ïd n = 1 ¿Ø´§ (6-3)  ). (ii).  n = p ž, Ù¥ 1 ≤ α ≤ k − 1, p ´ƒê. d¼ê S (n)  ½Â·‚k β 1 2 α 1 2 2 1 2 3 2 1 2 3 1 2 3 4 1 2 2 1 2 3 3 4 k k d|n k k d|n α X d|pα k Sk (d) = Sk (1) + Sk (p) + · · · + Sk (pα ) = α + 1. dž ω(n) = 1, Ω(n) = α,  ω(n)Ω(n) = α, X S (d) 6= ω(n)Ω(n), Ïd n = p ǑØ´§ (6-3) ). (iii). X n = p ·p · · · p ž, Ù¥ 1 ≤ α ≤ k−1, i = 1, 2, · · · , r, r ≥ 2, du S (d) ´˜‡Œ¼ê, ·‚k k d|n α α1 1 α2 2 αr r i k d|n X Sk (d) = α α r d|p1 1 p2 2 ···pα r Ӟ, X α d|p1 1 Sk (d) · X α d|p2 2 Sk (d) · · · X Sk (d) r d|pα r = (α1 + 1)(α2 + 1) · · · (αr + 1). ω(n) = r, Ω(n) = α1 + α2 + · · · + αr , ω(n)Ω(n) = r(α1 + α2 + · · · + αr ), (6-3) : dž§ Ǒ (α1 + 1)(α2 + 1) · · · (αr + 1) = r(α1 + α2 + · · · + αr ). (6-4) 95 'uSmarandache¯KïÄ#? e¡·‚lXeA«œ¹é¤kê r (r ≥ 2) ?1?Ø: X (a). e r = 2, S (d) = (α + 1)(α + 1), ω(n)Ω(n) = 2(α + α ), ·‚)§ (α + 1)(α + 1) = 2(α + α )  α = 1 ½ö α = 1. ¤ ± n = p p Ú n = p p ÷v (6-3), Ù¥ 1 ≤ α, β ≤ k − 1. (b). e r = 3, ÷v§ (6-4) ªǑ: k 1 2 1 2 d|n 1 2 β 1 2 α 1 2 1 2 1 2 (α1 + 1)(α2 + 1)(α3 + 1) = 3(α1 + α2 + α3 ). (6-5) e¡·‚é α Š?1?Ø, Ù¥ i = 1, 2, 3. i). e (6-5) ª¥k =k˜‡ α ÷v α = 1, ·‚Ø”- α 1, α , α > 1, džk i i 2 i 1 = 3 2(α2 + 1)(α3 + 1) = 3(1 + α2 + α3 ). du 2(α2 + 1)(α3 + 1) − 3(1 + α2 + α3 ) = 2α2 α3 − α2 − α3 − 1 = α2 (α3 − 1) + α3 (α2 − 1) − 1 > 0. ùÒy² 2(α + 1)(α + 1) o´Œu 3(1 + α + α ), Ïd§ (6-3) 3d«œ¹e ). ii). e (6-5) ª¥kü‡ α ÷v α = 1, ·‚Œ± α = α = 1, α > 1, )§ 4(α + 1) = 3(2 + α ),  α = 2, u´ n = p p p ½ö n = p p p ½ö n = p p p ÷v§ (6-3), ´§). iii). e¤k α Ñ÷v α = 1, § (6-5) ؤá, dž§ (6-3) ). iv). e¤k α ÷v α > 1, ·‚Œ±éN´y²e¡Øª ¤á: 2 3 2 i 3 3 2 1 2 3 i ÏǑ i 3 1 3 2 2 1 2 3 2 1 2 3 i = 3 i i (α1 + 1)(α2 + 1)(α3 + 1) > 3(α1 + α2 + α3 ), α1 α2 α3 + α1 α2 + α2 α3 + α1 α3 + 1 > 2α1 + 2α2 + 2α3 . (α1 α2 α3 + α1 α2 + α2 α3 + α1 α3 + 1) − (2α1 + 2α2 + 2α3 ) 96 18Ù ˜ ¹ Smarandache ¼ê§ = α1 α2 α3 + α1 (α2 − 2) + α2 (α3 − 2) + α3 (α1 − 2) + 1 ≥ α1 α2 α3 + 1 > 0. ùÒy² (6-5) ª†>o´Œum>, Ïd§ (6-3) 3ù«œ¹e Ǒ´ ). (c).  k = 4, ·‚k (α1 + 1)(α2 + 1)(α3 + 1)(α4 + 1) = 4(α1 + α2 + α3 + α4 ). (6-6) e¡·‚鐧 (6-6) ¥ α Š?1?Ø, Ù¥ i = 1, 2, 3, 4. i). § (6-6) ¥k k˜‡ α ÷v α = 1, ·‚Œ± α 1, α > 1, α > 1, α > 1, ù (6-6) ªCǑ i i 2 3 i 1 = 4 2(α2 + 1)(α3 + 1)(α4 + 1) = 4(1 + α2 + α3 + α4 ), = α2 α3 α4 + α2 α3 + α2 α4 + α3 α4 = 1 + α2 + α3 + α4 . ÏǑ α > 1, α > 1, α > 1, ù·‚Œ±éN´y² (6-6) ª† >o´Œum>, dž§ ). ii). § (6-6) ¥kü‡ α ÷v α = 1, ؔ- α = α = 1, α > 1, α > 1, ·‚k 2 3 4 i i 1 2 3 4 (α3 + 1)(α4 + 1) = 2 + α3 + α4 . w,ù‡ªØ¤á, Ïd3ù«œ¹e§ ). iii). § (6-6) ¥kn‡ α ÷v α = 1, Œ- α 1, α > 1, (6-6) ªCǑ i i 1 = α2 = α3 = 4 2(α4 + 1) = 3 + 4α4 . ¿ α = 1/2, ù´ØŒU, dž§ (6-3) ). iv). § (6-6) ¥¤k α ÷v α = 1, dž (6-6) ª¤á, Ï d n = p p p p ´§ (6-3) ). v). e¤ko‡ α þ÷v α > 1, du α > 1, α > 1 ž, (α + 1)(α + 1) > 2(α + α ). ÓnŒ α > 1, α > 1 ž, (α + 4 i i 1 2 3 4 i 1 2 1 i 2 1 3 2 4 3 1)(α4 + 1) > 2(α3 + α4 ). 97 'uSmarandache¯KïÄ#? u´·‚k (α1 + 1)(α2 + 1)(α3 + 1)(α4 + 1) > 4(α1 + α2 )(α3 + α4 ) = 4(α1 α3 + α1 α4 + α2 α3 + α2 α4 ) > 4(α1 + α2 + α3 + α4 ). Ïdª (6-6) †>o´Œum>, ¤±§ (6-3) 3d«œ¹e ). (d).  r > 4, 1 ≤ i ≤ r, Œ±y²§ (6-3) †>o´Œum >. = (α1 + 1)(α2 + 1) · · · (αr + 1) > r(α1 + α2 + · · · + αr ). ¤±§ (6-3) 3ù«œ¹e ). e¡·‚^êÆ8B{y²þãت¤á.  r = i ž, ت¤ á, = (α1 + 1)(α2 + 1) · · · (αi + 1) > i(α1 + α2 + · · · + αi ). K k = i + 1 ž, (α1 + 1)(α2 + 1) · · · (αi + 1)(αi+1 + 1) > i(α1 + α2 + · · · + αi )(αi+1 + 1). qÏǑ i(α1 + α2 + · · · + αi )(αi+1 + 1) − (i + 1)(α1 + α2 + · · · + αi + αi+1 ) = i(α1 + α2 + · · · + αi )(αi+1 + 1) − i(α1 + α2 + · · · + αi ) − iαi+1 −(α1 + α2 + · · · + αi ) − αi+1 = (iαi+1 − 1)(α1 + α2 + · · · + αi ) − (i + 1)αi+1 > (iαi+1 − 1)(α1 + α2 + · · · + αi ) − (i + 1)(iαi+1 − 1) > (iαi+1 − 1)(α1 + α2 + · · · + αi − i − 1) > 0. ¤±,  k = i + 1 ž, ت½¤á, l 3ù«œ¹e§ (6-3) ). (iv).  n = p Ú α ≥ k, dž·‚½Â α = kβ + γ  n = p , Ù¥ β ≥ 1, 0 ≤ γ < k, ·‚k α X d|pkβ+γ 98 Sk (d) kβ+γ 18Ù ˜ ¹ Smarandache ¼ê§ = Sk (1) + Sk (p) + · · · + Sk (pk−1 ) + Sk (pk ) + · · · + Sk (p2k−1 ) + Sk (p2k ) + · · · + Sk (pk(β−1)−1) + Sk (pk(β−1) ) + · · · + Sk (pkβ−1 ) + Sk (pkβ ) + Sk (pkβ+1 ) + · · · + Sk (pkβ+γ ) = |1 + ·{z · · + 1} + p + · · · + p + p2 + · · · + p2 + | {z } | {z } k k β−1 ··· + p | Ӟ, k β−1 + ··· + p {z k 2 } ÏǑ (γ + 1)p + p + · · · + pβ | {z } γ+1 β−1 = k(1 + p + p + · · · + p k(pβ − 1) = + (γ + 1)pβ . p−1 ω(n)Ω(n) = kβ + γ, β ) + (γ + 1)p⠐§ (6-3) Œ±CǑ: k(pβ − 1) + (γ + 1)pβ = kβ + γ. p−1 β > γ, ±9 k(pβ − 1) − (p − 1)kβ = kpβ − k − kpβ + kβ = k(pβ − pβ) + k(β − 1) > 0. u´·‚ k(pp −−11) + (γ + 1)p > kβ + γ, dž§ ). (v). e n = p p · · · p Ú α ≥ k, i = 1, 2, · · · , r, ·‚- α = kβ + γ , β ≥ 1, 0 ≤ γ < k  n = p p ···p , Ï Ǒ S (n) ´˜‡Œ¼ê, ·‚k β β α1 α2 1 2 i i i αr r i i kβ1 +γ1 kβ2 +γ2 1 2 i kβr +γr r k X Sk (d) α α r d|p1 1 p2 2 ···pα r = X Sk (d) kβ1 +γ1 kβ2 +γ2 p2 ···prkβr +γr d|p1 = X d|pkβ1 +γ1  Sk (d) · X d|pkβ2 +γ2 Sk (d) · · · X Sk (d) d|pkβr +γr    k(pβ1 − 1) k(pβ2 − 1) β1 β2 = · + (γ1 + 1)p + (γ2 + 1)p p−1 p−1   k(pβr − 1) βr · ··· · . + (γr + 1)p p−1 99 'uSmarandache¯KïÄ#? Ӟ, ω(n)Ω(n) = r [k(β Ïd§ŒCǑ 1  k(pβ1 −1) p−1 + β2 + · · · + βr ) + γ1 + γ2 + · · · + γr ] .   β  2 + (γ1 + 1)pβ1 · k(pp−1−1) + (γ2 + 1)pβ2 · · · ·   βr k(p −1) βr + (γr + 1)p · p−1 = r [k(β1 + β2 + · · · + βr ) + γ1 + γ2 + · · · + γr ] . (6-7) Œ±y² (6-7) ª†>o´Œum>. ·‚|^êÆ8B{?1y²:  r = i, ت¤á, =  k(pβ1 −1) p−1   β  2 + (γ1 + 1)pβ1 · k(pp−1−1) + (γ2 + 1)pβ2 · · · ·   β k(p i −1) βi + (γi + 1)p · p−1 > i [k(β1 + β2 + · · · + βi ) + γ1 + γ2 + · · · + γi ] . K r = i + 1 ž,    β  2 + (γ1 + 1)pβ1 · k(pp−1−1) + (γ2 + 1)pβ2 · · · ·   βi+1   β i · k(pp−1−1) + (γi + 1)pβi · k(p p−1−1) + (γi+1 + 1)pβi+1  βi+1  k(p −1) βi+1 > i [k(β1 + β2 + · · · + βi ) + γ1 + γ2 + · · · + γi ] + (γi+1 + 1)p p−1 k(pβ1 −1) p−1 > (i + 1) [k(β1 + β2 + · · · + βi+1 ) + γ1 + γ2 + · · · + γi+1 ] . ¤± k = i + 1 ž, ت¤á. (vi). e n = p p · · · p p p ···p , Ù¥ α > k (i = 1, 2, · · · , r), 1 ≤ α < k (r + 1 ≤ j ≤ r + t), l±þ?،, d «œ¹e§ (6-3) ). nܱþ?Ø, ·‚¤ ½ny². α1 α2 1 2 αr αr+1 αr+2 r r+1 r+2 αr+t r+t i j 6.5 'u Smarandache ¯K˜‡í2 ؽ§ (½§|) ´Cþê‡êõu§‡ê, CþêŠ § (½§|). ؽ§´êØ¥P q­‡˜‡©|, ~X Ͷ Fermat Œ½nҴؽ§˜‡;.“L, ÙSN†y“ê 100 18Ù ˜ ¹ Smarandache ¼ê§ Æk—ƒéX. F. Smarandache Ç3©z [1] 1 50 ‡¯K¥ïÆ· ‚ïЧ 1 1 xa x + ax = 2a x . (6-8) Œ)5, ¿T§¤k¢ê) 'uù˜¯K, Ü©+Ç3©z [56] ¥?1 ïÄ. äN/`, = y²e¡(Ø: ½n. é¤k a ∈ R\{−1, 0, 1}, § (6-8) k =k˜‡¢ê ) x = 1. !ò§ (6-8) ?1 í2Úò, =Ä n − 1 ‡Cþœ ¹, Ñ § 1 1 1 x1 a x1 + x2 a x2 + · · · + xn−1 a xn−1 + 1 ax1 x2 ···xn−1 = na (6-9) x1 x2 · · · xn−1 ¤kšK¢ê), ½=y² e¡½n. ½n 6.9. é?¿¢ê a ∈ R\{0}, § (6-9) k =k˜|šK ¢ê) [59] x1 = x2 = · · · = xn−1 = 1.  n = 2 ž, § (6-9) CǑ x a + a = 2a, =©z [56] ¥¤ ?Øœ¹. Ïd©(J´©z [56] ¥½ní2Úò. y²: |^©z [56] ¥gŽ†‰Ñ½ny². ǑdI‡e¡ ü‡(Ø: (Ø 1. é?¿¢ê a , a , · · · , a , kت 1 1 2 1 x1 1 x1 x1 n √ a1 + a2 + · · · + an ≥ n a1 a2 · · · an . n õ¼ê4Š7‡^‡: ¼ê z = f (x , x , · · · , x ) 3 : (x , x , · · · , x ) äk ê 4Š, K§3T: ê7Ǒ", = ( Ø 2. f (x , x , · · · , x ) = f (x , x , · · · , x ) = · · · = 1 ′ 1 ′ 2 ′ n n x1 ′ 2 ′ ′ 1 ′ 2 ′ n x2 ′ 1 ′ 2 ′ n ′ fxn (x1 , x2 , · · · , xn ) = 0. 101 'uSmarandache¯KïÄ#? ±þü‡(Øy²Œë©z [9]![57] 9 [58]. y3|^±þü‡(Ø5y²é¤k a ∈ R\{0}, § 1 1 ax1 x2 ···xn−1 = na x1 x2 · · · xn−1 1 1 x1 a x1 + x2 a x2 + · · · + xn−1 a xn−1 + ¤á,  = x = x = · · · = x = 1. y3ò a ©¤n«œ ¹ a ≥ 1, 0 < a < 1 9 a < 0 ?Ø. ¯¢þ,  a ≥ 1 ž, dþ¡ÄتŒ 1 2 n−1 1 1 1 + + ··· + + x1 x2 · · · xn−1 ≥ a. x1 x2 xn−1 u´k 1 1 1 x1 a x1 + x2 a x2 + · · · + xn−1 a xn−1 + q ≥n n 1 a x1 þª¤á = x + x1 +···+ x 1 2 1 +x1 x2 ···xn−1 n−1 √ ≥ n n an ≥ na, = x2 = · · · = xn−1 =  0 < a < 1 ž, ǑBå„,  x 1 n = ½ö x = =k˜|¢ê 1 x1 x2 · · · xn−1 (6-9) Q a ≥ 1 ž, § x2 = · · · = xn−1 = 1. x1 = x2 = · · · = xn−1 = 1. ) 1 x1 x2 ···xn−1 x1 x2 ···xn−1 a 1 , x1 x2 · · · xn−1 1 1 ¿-¼ê 1 f (x1 , x2 , · · · , xn−1 ) = x1 a x1 + x2 a x2 + · · · + xn−1 a xn−1 1 + ax1 x2 ···xn−1 − na, x1 x2 · · · xn−1 Ï Ǒ ¼ ê f (x , x , · · · , x ) 3 (0, +∞) þ ´ Œ  ,  é f (x , x , · · · , x ) ©O'u x , x , · · · , x  ê, Œ 1 1 2 2 n−1 1 ∂f ln a = a x1 (1 − )+ ∂x1 x1 1 ln a = a x1 (1 − )+ x1 1 ∂f ln a = a x2 (1 − )+ ∂x2 x2 102 n−1 1 2 n−1 1 1 1 ln aax1 x2 ···xn−1 − ax1 x2 ···xn−1 x1 x1 x1 x2 · · · xn−1 1 x1 a n (ln a − xn ), x1 1 x1 a n (ln a − xn ), x2 18Ù ˜ ¹ Smarandache ¼ê§ ··· 1 ln a 1 x1 ∂f = a xn−1 (1 − )+ a n (ln a − xn ). ∂xn−1 xn−1 xn−1 ©O- ∂x∂f , ∂x∂f , · · · , ∂x∂f 1 2 = 0, n−1 K 1 1 1 1 a x1 (ln a − x1 ) = a xn (ln a − xn ), a x2 (ln a − x2 ) = a xn (ln a − xn ), a 1 xn−1 ··· 1 (ln a − xn−1 ) = a xn (ln a − xn ). ¼ê u(x) = a (ln a − x), Ù¥ 0 < a < 1. e¡y²d¼êǑüN¼ ê. 1 x ′ 1 ln a(x − ln a) − 1] x2 1 ln a 2 ln a ) − + 1] = −a x [( x x 1 ln a 1 2 3 − ) + ] < 0. = −a x [( x 2 4 1 u (x) = a x [  u(x) 3 (0, +∞) þ´üN4~¼ê, ¤±éAuӘ‡¼êŠ u(x ), k =k˜‡ x ¤á. ¤±d 0 0 1 1 1 a x1 (ln a − x1 ) = a x2 (ln a − x2 ) = · · · = a xn (ln a − xn) Œ  Ñ x = x = · · · = x = q. K (q, q, · · · , q) Ǒ  ¼ ê f (x , x , · · · , x ) ŒU4Š:, =kù˜‡4Š:. d q = 1 Œ q = 1,  (1, 1, · · · , 1) Ǒ¼ê4Š:, dž4ŠǑ 0. ¯¢þ, b ¼ê f (x , x , · · · , x ), „kÙ4Š:, Ǒ (x , x , · · · , x ), Kdõ¼ê4Š37‡^‡Œ, 3ù˜:7k x = x = · · · = = 1 ª¤á, †ƒ üN¼êӘ¼êŠéA˜gCþgñ.  x ¼êk: (1, 1, · · · , 1) ù˜‡4Š:. =dž¼ê 1 1 2 2 n n n−1 1 2 ′ ′ ′ n−1 n−1 2 1 ′ ′ 1 2 ′ n−1 1 1 1 f (x1 , x2 , · · · , xn−1 ) = x1 a x1 + x2 a x2 + · · · + xn−1 a xn−1 1 ax1 x2 ···xn−1 − na, + x1 x2 · · · xn−1 103 'uSmarandache¯KïÄ#? ˜4Š, 4ŠǑ 0. Q = x = x = · · · = x §)´ 1 2 n−1 =1 ž,  x1 = x2 = · · · = xn−1 = 1.  a < 0 ž, § (6-9) e‡k), K xp , x , · · · , x 7Ǒkn ê, ÏǑKêØUm nêg,  x = q , (p , q ) = 1, Ù¥ i = 1, 2, · · · , n. e‡ a k¿Â, K p 7ǑÛê,  p p · · · p ǑǑÛê, e 3 q Ǒóê, Kd®^‡k pq ·· ·· ·· pq ·· ·· ·· qp = 1, ù† p ǑÛêgñ. ¤± q q · · · q ǑǑÛê, q ǑÛê, ¤±k a = −|a| , u´§Œ zǑ 1 i i qi pi i i 1 2 i n 2 n−1 i i 1 2 1 i n 1 i n 1 xi i n i 1 xi 1 ax1 x2 ···xn−1 x1 x2 · · · xn−1 1 + |a|x1 x2 ···xn−1 . x1 x2 · · · xn−1 1 1 n|a| = −na = −x1 a x1 − · · · − xn−1 a xn−1 − 1 1 = x1 |a| x1 + · · · + xn−1 |a| xn−1 dž, d ¡?Øü«œ¹Œ, § (6-9) )E,´ x1 = x2 = · · · = xn−1 = 1. nþ¤ã, Œѐ§ (6-9) ¤kšK¢ê)Ǒ x1 = x2 = · · · = xn−1 = 1. u´, ¤ ½ny². 104 1ÔÙ 1ÔÙ 7.1 Smarandache ¼êƒ'¯K Smarandache Smarandache ¼êƒ'¯K ¼ê·ÜþŠ¯K 3 ¡®²0 L Smarandache ¼ê S(n) 9ÏfÈê {P (n)} ½Â, = d S(n) = min{m : m ∈ N, n | m} Pd (n) = n d(n) 2 . 'uùü‡¼êˆ«5Ÿ, ®kNõ<?1ïÄ. ֊ö3 <ïĤJþE¿)û ˜‡#·ÜþŠ¯K („©z [60]), äN`Ò´|^9)ې{ïÄ ·ÜþŠ 2 X 1 S (Pd (n)) − d (n) P (n) 2 n≤x ìC5Ÿ§¿‰Ñ ˜‡rìCúª. ½n 7.1.  N ≥ 1 Ǒ‰½ê. éu?¿¢ê x > 1, ·‚k ìCúª X n6x 2 X 3 N x2 1 ci · i + O S (Pd (n)) − d (n) P (n) = 2 ln x i=1 3 x2 lnN+1 x Ù¥ c (i = 1, 2, · · · , N) ´ŒOŽ~ê Ǒ Riemann zeta- ¼ê. Ǒ y²ù‡(Ø, k‰ÑA‡Ún. Ún 7.1.1.  n ≥ 1 Ǒê√, K (i) XJ n k˜‡ƒÏf p > n§K S (P (n)) = √ (ii) XJ n = n p p n < p < p 6 n§K ·p ; (iii) XJ n = n p p > n , Kk S (P (n)) − ,
3 ζ 4 23 , ζ(n) c1 = · 2 ζ(3) i d(n) 2 d 1 1 2 d(n) 2 ! 1 3 1 2 · p; S (Pd (n)) = 2 1 3 pd (n1 ) . 2 2 1 d 1 d (n) P 2 (n) = 105 'uSmarandache¯KïÄ#? y ²: é ? ¿   ê n,  n = p p · · · p Ǒ n  I O © ) ª. u ´ d Smarandache ¼ ê  5 Ÿ Œ  S(n) = max{S (p ) , S (p ) , · · · , S (p )}. u´ √ 1 (i).  n k˜‡ƒÏf p > n ž, 5¿dž p ≥ d(n), 2 d Smarandache ¼ê5Ÿ9®^‡k αk k α1 α2 1 2 α1 1 αk k α2 2  d(n)   d(n)  d (n) · p. S (Pd (n)) = S n 2 = S p 2 = 2 √ 1 (ii). n = n1 p1 p2 n 3 < p1 < p2 6 n , Smarandache S (m1 p1 p2 ) = p2 ,  d(n)   d(n) d(n) d(n)  d(n) 2 2 2 = S p2 2 = · p2 . S (Pd (n)) = S m1 p1 p2 2  ê5ŸŒ d  žd ¤± ¼ ž, džw,k n < p 6 √n. u´ ¼ê5Ÿ¿5¿ d(n) = 3d(n ) ·‚k: n = n1 p2 (iii). Smarandache 1 3 p > n1 1
3 1 d (n) 1 1 S (Pd (n))− d (n) P (n) = 2p· − pd (n) = pd n1 p2 = p·d (n1 ) . 2 2 2 2 2 u´y² Ún 7.1.1. Ún 7.1.2. - p Ǒƒê, m Ǒê, 3 N 2 x 2 X ci · lni−1 m +O p = 3 m 23 i=1 lni x √x X m6p6 1 2 3 m 6 x3 , ! 3 x2 m 2 lnN+1 x KkìCúª +O m   m3 , ln m Ù¥ c ǑŒOŽ~ê c = 1. y²: w,dƒê½n·‚k² Oª: i 1 X p6m 2 2 p = m π(m) − Z m 2yπ(y)dy = O 2 u´d©z [3] ±9 Abel Úúª·‚áǑíÑ: X √x m6p6 106 m p2 = X √x p6 m p2 + O  m3 ln m  x = ·π m  r m3 ln m x m   − . Z √x m 2 2yπ(y)dy 1ÔÙ Smarandache ¼êƒ'¯K 3 N 2 x 2 X ci · lni−1 m = +O 3 m 23 i=1 lni x 3 x2 3 m 2 lnN+1 x ! +O  m3 ln m  , Ù¥ c ǑŒOŽ~ê c = 1. u´¤ Ún 7.1.2 y². Ún 7.1.3. é?¿¢ê x > 3,  A L«¤kùê n 8 ܵé?¿ƒê p, p|n  = p ≤ n . KkXeOª: i 1 1 3 2 X 4 1 S (Pd (n)) − d (n) P (n) ≪ x 3 ln2 x. 2 n6x n∈A y²: 5¿é?¿ê n ∈ A§w,d Smarandache ¼ê S(n) 1 1 5Ÿ p|n ž, XJ S(n) = p, K S (P (n)) = 2 d(n)P (n) = 2 d(n)·p, džk   d 1 S (Pd (n)) − d (n) P (n) 2 XJ S (P (n)) 6= 12 d(n)P (n),  Oª: X = 0. S(n) = S(pα ). d n≤M 2 Kw,k α ≥ 2. 5¿ d2 (n) ≪ M · ln3 M. u´dƒê½n (ë©z [8] 9 [27]) ·‚k 2 X X 1 p2 d2 (n) S (Pd (n)) − d (n) P (n) ≪ 2 2 n6x n∈A ≪ X 1 p≤x 3 np ≤x p≤n p2 X 4 p<n≤ px2 d2 (n) ≪ x 3 ln2 x. l y² Ún 7.1.3. ½ny²: ŠâÚn 7.1.1  (i) ªé?¿ê n, XJ3 · P (n) = 0. u´(ÜÚn 7.1.1 ƒê p|n p > √n, K S (P (n)) − d (n) 2  (ii) ªÚÚn 7.1.3 ¿5¿ Riemann zeta- ¼ê𪵠d ∞ X d2 (n) 3 n=1 n2
ζ 4 23 = , ζ(3) 107 'uSmarandache¯KïÄ#? ·‚áǑ X 2 1 S (Pd (n)) − d (n) P (n) 2 n6x 2 X  1 = S (Pd (n)) − d (n) P (n) + 2 n6x √ p|n, p> n X (S (Pd (n)) n6x √ p|n, p6 n 2 1 − d (n) P (n) 2 2 X  1 S (Pd (n)) − d (n) P (n) = 2 n6x √ p|n, p6 n X = n6x 1 2  1 S (Pd (n)) − d (n) P (n) + 2 p|n, p6n 3 X (S (Pd (n)) n6x 1 √ p|n, n 3 <p6 n 2 1 − d (n) P (n) 2   4  X 

2

1 2 2 2 2 3 + O x ln x = S Pd n1 p − d n1 p P n1 p 2 2 n1 p 6x n1 <p X = 1 X n6x 3 n<p≤ √x n 9 X 2 d (n) = 4 1 n6x 3 N X 3 2  3 pd (n) 2 X n<p≤ x = ci · i + O ln x i=1 2  4  + O x 3 ln2 x  4  2 3 p + O x ln x 2 √x n 3 x2 lnN+1 x ! , Ù¥ c (i = 1, 2, · · · , N) ´ŒOŽ~ê l ¤ ½ny². ½n3Ͷ Smarandache ¼ê S(n) 9ÏfÈê {P (n)} Ä :þ, E ˜‡Žâ¼ê†ŒƒÏf¼ê¿|^{Úƒê½ nïÄ §‚·ÜþŠ¯K, ¿‰Ñ §‚˜‡rì?úª. i
3 ζ 4 23 . c1 = · 2 ζ(3) d 108 1ÔÙ Smarandache ¼êƒ'¯K k,ÖöŒ±æaq{, 5Eõ¼ê, ¿ïħ‚·Ü þŠ. 'u²Öê SSC(n) ü‡¯K 7.2 F. Smarandache : Ç3©z [1] ¥„JÑ ±eù‡ Smarandache ²Öê¼ê ½Â 7.1. é?¿ê n, Ͷ Smarandache ²Öê¼ ê SSC(n) ½ÂǑê m  m · n Ǒ²ê, =Ò´ SSC(n) = min{m : m · n = k 2 , m, k ∈ Z ∗ }. d ½Â·‚ØJOŽÑ  A ‡ Š Ǒ: SSC(n) SSC(n) SSC(1) = 1, SSC(2) = 2, SSC(3) = 3, SSC(4) = 1, SSC(5) 5, SSC(6) = 6, SSC(7) = 7, SSC(8) = 2, SSC(9) = 1, SSC(10) 10, SSC(11) = 11, SSC(12) = 3, SSC(13) = 13, SSC(14) 14, SSC(15) = 15, SSC(16) = 1, SSC(17) = 17, SSC(18) 2, SSC(19) = 19, SSC(20) = 5, · · · . = = = = 'u SSC(n) 5Ÿ, NõÆö?1 ïÄ,  éõkd Š¤J. ~X, Russo [61] é SSC(n) ?1 ïÄ, Ñ 'u SSC(n) ˜ 5Ÿ: 5Ÿ 1. éu?¿ê n, k SSC(n) ≤ n. 5Ÿ 2. éu?¿ê n, XJ n IO©)ªǑ n = p p · · · p , o α1 α2 1 2 αs s odd(α1 ) odd(α2 ) p2 SSC(n) = p1 Ù¥ α ≥ 0, p i i s) · · · podd(α , s ´p؃Óƒê, ¼ê odd(n) ½ÂǑ: ( 1, en´Ûê; odd(n) = 0, en´óê. (i = 1, 2, · · · , s) 109 'uSmarandache¯KïÄ#? Russo ӞJÑXe¯K: n X ln SSC(k) ¯K 1. OŽ4 n→∞ ¯K 2. OŽ4 SSC(n) , n→∞ θ(n) lim k=2 ln k n lim . Ù¥ θ(n) = P ln SSC(k). k≤n 'uùü‡¯K, –8qvk<ïÄ, –·‚vkwLk' ¡Ø©. CRŸ [62] )û ùü‡¯K, Ñ ±e(Ø: ½n 7.2. éu?¿ê n ≥ 1, kOª n ln SSC(k) P   ln k 1 k=2 . =1+O n ln n íØ 7.2.1. éu?¿ê n ≥ 1, k4 n ln SSC(k) P ln k = 1. lim k=2 n→∞ n ½n 7.3. éu?¿ê n ≥ 1, kOª SSC(n) =O θ(n)  1 ln n  . íØ 7.2.2. éu?¿ê n ≥ 1, k4 lim n→∞ SSC(n) = 0. θ(n) Ǒ ¤½ny², I‡±eA‡Ún: 110 1ÔÙ Smarandache ¼êƒ'¯K éu?¿¢ê x ≥ 2, kìCúª Ún 7.2.1. X µ2 (n) = n≤x √
6 x . x + O π2 (7-1) 5¿ ζ(2) = π6 , Ïd (7-1) ªŒǑ 2 X µ2 (n) = n≤x √
x +O x , ζ(2) (7-2) Ù¥ ζ(n) Ǒ Riemann zeta- ¼ê. Ún 7.2.2. éu?¿¢ê x ≥ 2, kìCúª X n≤x ln SSC(n) = x ln x − Ax + O
√ x ln2 x , (7-3) Ù¥ A Ǒ~ê. y²: XJ^ A L«¤k ²Ïfê8Ü, od Abel Ú úª (ë©z [8] ¥½n 4.2!Ún 1 95Ÿ 2), k X ln SSC(n) n≤x X = ln SSC(m2 l) = m2 l≤xl∈A = X X √ m≤ x l≤ mx2 X ln l m2 l≤xl∈A ln l · µ2 (l) )   √  Z x  √  m2 1 x x t 1 x t dt ln 2 = − +O +O m ζ(2) m2 m t ζ(2) √ 1 m≤ x  √ X  x ln x 1 x ln x 2x ln m x 1 = . (7-4) − − +O ζ(2) m2 ζ(2) m2 ζ(2) m2 m √ X ( m≤ x 5¿eA‡ìCúª [8]   X1 1 , = ln x + C + O n x n≤x : Ù¥C Ǒ~ê; 111 'uSmarandache¯KïÄ#? Z ∞ 1   X 1 1 1 = ζ(2) − + O ; 2 n x x2 n≤x   X ln n ln x 1 ln x = B − − + O , n2 x x x2 n≤x Ù¥B = 1 + Ǒ~ê. (t − [t])(t − 2t ln t) dt t4 (7-4) , d ªk X n≤x ln SSC(n) = x ln x X 1 2x X ln m − 2 ζ(2) ζ(2) √ m2 √ m m≤ x m≤ x
√ x X 1 2 x ln x − + O ζ(2) √ m2 m≤ x
√ 2 Bx − x + O x ln2 x ζ(2)
√ = x ln x − Ax + O x ln2 x , = x ln x − 2B Ù¥ A = ζ(2) + 1 Ǒ~ê. u´¤ Ún 7.2.2 y². ½ny²: 3ù˜Ü©, ^{‰Ñ½ny². Äky² ½n 7.2. ˜¡, d5Ÿ 1, k n ln SSC(k) n ln k P P ln k ln k n−1 k=2 ≤ k=2 = < 1. n n n (7-5) ,˜¡, dÚn 7.2.2, k n ln SSC(k) P ln k k=2 n 112 n 1 X ln SSC(k) > n ln n k=2   ln n A +O √ = 1− ln n n   1 = 1+O . ln n (7-6) 1ÔÙ Smarandache ¼êƒ'¯K (Ü (7-5)!(7-6) ª, k n ln SSC(k) P   ln k 1 k=2 . =1+O n ln n u´¤ ½n 7.2 y². íØ 7.2.1 Œn)Ǒ½n 7.2 ¥ n → ∞ ž4. y3y²½n 7.3. d5Ÿ 1!Ún 7.2.2 9 SSC(n) ½Â, k n SSC(n) < =O 0< θ(n) θ(n) d (7-7) ª, k SSC(n) =O θ(n)  1 ln n   1 . ln n  . (7-7) u´y² ½n 7.3. íØ 7.2.2 Œn)Ǒ½n 7.3 ¥ n → ∞ ž4. ±þ·‚ÏL|^9)ې{ïÄ ln SSC(n) Š©Ù5 Ÿ, l ò Russo [61] JÑü‡4¯K”.)û. 'u Smarandache ²Öê¼ê SSC(n) 5Ÿ8 ƒ$, „kNõ¯Kk–k, Æö?1ïÄ. ~X, ¯K 1. § S(n) + Z(n) = SSC(n) ¤kê). ¯K 2. ïļê SSC(Z(n)) 9¼ê Z(SSC(n)) 5Ÿ. k 7.3 k k 'u Smarandache ˜¼ê˜‡þŠ¯K ½Â 7.2. é?¿ê n, Ͷ Smarandache ˜¼ê SP (n) ½ ÂǑê m  n|m , Ù¥ m Ú n kƒÓƒÏf. =µ m    Y Y  SP (n) = min m : n|mm , m ∈ N, p= p ,   p|n p|m 113 'uSmarandache¯KïÄ#? Ù¥ N L«¤kg,ê8Ü. ~X, SP (n)  A‘Ǒ: SP (1) = 1, SP (2) = 2, SP (3) = 3, SP (4) = 2, SP (5) = 5, SP (6) = 6, SP (7) = 7, SP (8) = 4, SP (9) = 3, SP (10) = 10, SP (11) = 11, SP (12) = 6, SP (13) = 13, SP (14) = 14, SP (15) = 15, SP (16) = 4, SP (17) = 17, SP (18) = 6, SP (19) = 19, SP (20) = 10, · · · . Ç3©z [1] ¥ïÆ·‚ïÄ SP (n) 5Ÿ. ½Â éN´e¡(Ø: e n = p , Ù¥ p Ǒ˜‡ƒ F. Smarandache SP (n) , l ê, Kk   p,     2   p , SP (n) = p3 ,    ···     pα , α 1 ≤ α ≤ p; p + 1 ≤ α ≤ 2p2 ; 2p2 + 1 ≤ α ≤ 3p3 ; ··· (α − 1)pα−1 + 1 ≤ α ≤ αpα . - n = p p · · · p L« n ƒê˜©)ª. w,, SP (n) Ø´Œ¼ê. ~X, SP (8) = 4, SP (3) = 3, SP (24) = 6 6= SP (3) × SP (8). éA¤k m Ú n (m, n) = 1, Ñk SP (mn) = SP (m) · SP (n). 3©z [63] ¥, M󸮲ïÄ SP (n) þŠ5Ÿ, ¿¼ ˜ ‡rìCúªµ α1 α2 1 2 αr r    3  1 2Y 1 SP (n) = x + O x 2 +ǫ , 1− 2 p(p + 1) p n≤x X Ù¥ ǫ L«?¿½ê, Y p L«é¤kƒê p È. ïÄ ˜‡¹ SP (n)  ¡?êÂñ5¯ K, ¿y² é?¿Eê s ÷v Re(s) > 1, k Zhou Huanqin [64] 114  s 2 +1 1   , k = 1, 2;  s − 1 ζ(s)  2  ∞  2s + 1 1 X 2s − 1 (−1)µ(n) − , k = 3; = k ))s 2s − 1 ζ(s) 4s  (SP (n  n=1   2s − 1 3s − 1 2s + 1 1   s − + , k = 4, 5. 2 − 1 ζ(s) 4s 9s 1ÔÙ Smarandache ¼êƒ'¯K X¥ 3©z [65] ¥|^{ïÄ SP (n) †²;êؼê î.¼ê φ(n) 'X, =ïÄ § SP (n ) = φ(n) Œ)5, ¿‰Ñ k = 1, 2, 3 ž¤kê), =e¡(Ø: (1) SP (n) = φ(n) k =k 4 ‡ê) n = 1, 4, 8, 18. (2) § SP (n ) = φ(n) k =k 3 ‡ê) n = 1, 8, 18. (3) § SP (n ) = φ(n) k =k 2 ‡ê) n = 1, 16. k 2 3 ܩ̇8´|^)ې{ïÄ SP (n)  k g˜©Ù5 P P (k > 0, l ≥ 0) ìCúª, í Ÿ, ‰Ñ n (SP (n)) 9 2 ©z [63] (Ø. ½n 7.4. é?¿¢ê x ≥ 3 9‰½¢ê k, l (k > 0, l ≥ 0), kìCúª l k n≤x n≤x (SP (n))k nl [67]    Y 1 ζ(k + 1) 1 k+l+1 n (SP (n)) = +O xk+l+ 2 +ε , x 1− k (k + l + 1)ζ(2) p (1 + p) p n≤x X l k X (SP (n))k n≤x nl Q    Y 1 ζ(k + 1) 1 k−l+1 +O xk−l+ 2 +ε , = x 1− k (k − l + 1)ζ(2) p (1 + p) p Ù¥, L«é¤kƒê p È, ε L«?¿ê, ζ(s) L« Riemann zeta- ¼ê. íØ 7.3.1. é?¿¢ê x ≥ 3 9‰½¢ê k > 0, kìCúª p ′ X (SP (n)) n≤x AO/, 1 ′ k ′ ′ ′ k +1 Y 6k ζ( 1+k ′ ) 1 k x k′ = ′ 1− 1 2 (k + 1)π p k′ (1 + p) p 9ζ( 43 ) 4 Y 1 x3 (SP (n)) = 1− 1 2 2π p 3 (1 + p) p n≤x X 1 3 ! 4ζ( 23 ) 3 Y 1 x2 (SP (n)) = 1− 1 2 π p 2 (1 + p) p n≤x X 1 2 ! !  +O x ′ k +2 ′ +ε 2k  ,  5  + O x 6 +ε ,
+ O x1+ε . 115 'uSmarandache¯KïÄ#? é?¿¢ê x ≥ 3 9‰½¢ê, kìCúª íØ 7.3.2.   3  Y 1 1 l+2 n (SP (n)) = + O xl+ 2 +ε , x 1− (l + 2) p(1 + p) p n≤x X l    5  6ζ(3) l+3 Y 1 n (SP (n)) = + O xl+ 2 +ε , x 1 − 2 (1 + p) (l + 3)π 2 p p n≤x   7  Y X 1 π2 l+4 l 3 + O xl+ 2 +ε . x 1− 3 n (SP (n)) = 15(l + 4) p (1 + p) p n≤x X l 2 Ǒ ¤½ny², I‡eÚn.  s = σ + it, ζ(s) Ǒ Riemann zeta- ¼ê, kQ> 0, l ≥ 0 Ǒ‰½ü ‡¢ê, p Ǒƒê. - n = p p · · · p , U(n) = p. α1 α2 1 2 αr r p|n Ún 7.3.1. é?¿¢ê x ≥ 1 9‰½¢ê k ≥ 1, kìCúª X n≤x     1 ζ(k + 1) k+1 Y k+ 21 +ε +O x x 1− k (U(n)) = . (k + 1)ζ(2) p (1 + p) p k , du U(n) ´È5¼ê, Šâ©z [66] y²: - A(t) = P (U(n)) n ¥ Euler È©ª,  σ > k + 1 ž, Œ k ∞ n=1 s # " ∞ m k Y X (U(p )) = A(s) = s n pm p m=0 n=1    Y 1 pk ζ(s)ζ(s − k) Y pk 1− k . = 1 + s + 2s + · · · = s−k ) p p ζ(2s − 2k) p (1 + p p p ∞ X (U(n))k - h(s) = ∞ X (U(n))k n=1 nσ 116 p  1 , 1− k p (1 + ps−k ) < ζ(σ − k), X a(n) n≤x Y ns0  σ > k + 1 ž, |U(n)| ≤ n, d©z [66] ¥ Perron úª, 1ÔÙ Smarandache ¼êƒ'¯K   b x B(b + σ0 ) xs A(s + s0 ) ds + O s T b−iT     lg x x 1−σ0 −σ0 +O x H(2x) min(1, ) + O x H(N) min(1, ) , T T kxk 1 = 2πi Z b+iT Ù¥, N Ǒl x Cê,  x ǑŒÛêž,  N = x − 21 , kxk = 3 , H(x) = |x − N|.  a(n) = (U(n)) , s = 0, b = k + , T = x 2 x, B(σ) = ζ(σ − k), K k 1 (U(n)) = 2πi n≤x X k òÈ©‚l Z k+ 32 +iT k+ 23 −iT s   ζ(s)ζ(s − k) xs k+ 21 +ε . h(s) + O x ζ(2s − 2k) s £  k + 12 ± iT , d ž ¼ 3 s = k + 1 ?k˜‡˜4:, Ù3êǑ s = k + − k) x ê ζ(s)ζ(s h(s) ζ(2s − 2k) s k+ 21 0 3 ± iT 2  xs ζ(s)ζ(s − k) h(s) L(x) = Res s=k+1 ζ(2s − 2k) s   ζ(s)ζ(s − k) xs = lim (s − k − 1) h(s) s→k+1 ζ(2s − 2k) s ζ(k + 1) k+1 = x h(k) (k + 1)ζ(2)  Y 1 . , h(k) = 1− k , p (1 + p) p Z k+ 1 +iT Z k+ 1 −iT Z k+ 3 +iT ! 2 2 2 1 ζ(s)ζ(s − k) 1 xs + + h(s) ≪ xk+ 2 +ε . 2πi ζ(2s − 2k) s k+ 23 +iT k+ 21 +iT k+ 21 −iT  Ù¥ N´O ¤±,     1 1 ζ(k + 1) k+1 Y + O xk+ 2 +ε . x 1− k (U(n)) = (k + 1)ζ(2) p (1 + p) p n≤x X k Ún 7.3.1 y. Ún 7.3.2. é?¿¢ê x ≥ 3!‰½¢ê k > 0 9ê α, k X pα ≤x α>p (αp)k ≪ ln2k+1 x. 117 'uSmarandache¯KïÄ#? y²:  π(x) = P 1, d©z [27] , p≤x x +O π(x) = ln x d Abel ª, Œ x p = π(x)x − k
lnk x + O lnk−1 x − k = (k + 1) Z ln x p≤x  p≤ln x k p x ln2 x Z X X  k k . π(t)tk−1 dt. 2 2 Z tk dt + O ln t ln x 2  tk dt ln2 x
lnk x + O lnk−1 x . k+1 = ÏǑ α > p, ¤± p p < pα ≤ x. p< q X o ln x ln x < ln x, α ≤ . ln p ln p nk = n≤x l X  (αp)k pα ≤x α>p = X pk p≤ln x ≪ lnk+1 x
xk+1 + O xk . k+1 X x α≤ ln ln p X p≤ln x αk ≪ lnk+1 x X p≤ln x pk lnk+1 p pk ≪ ln2k+1 x. Ún 7.3.2 y. ½ny²: - A = {n|n = p p · · · p , α ≤ p , i = 1, 2, · · · , r},  n ∈ A ž, k SP (n) = U(n),  n ∈ N ž, k SP (n) ≥ U(n), l α1 α2 1 2 X (SP (n))k − n≤x 118 X (U(n))k = n≤x αr r i i X  (SP (n))k − (U(n))k ≪ n≤x X n≤x SP (n)>U(n) (SP (n))k . 1ÔÙ Smarandache ¼êƒ'¯K d©z [63] , 3ê α 9ƒê p,  SP (n) < αp, ŠâÚ n 7.3.2 Œ X X (SP (n))k < n≤x SP (n)>U(n) n≤x SP (n)>U(n)  X n≤x (SP (n))k − dÚn 7.3.1 , X (αp)k ≪ X XX n≤x pα ≤x α>p (αp)k ≪ x ln2k+1 x. (U(n))k ≪ x ln2k+1 x. n≤x (SP (n))k n≤x    
1 ζ(k + 1) k+1 Y 1 = + O xk+ 2 +ε + O x ln2k+1 x x 1− k (k + 1)ζ(2) p (1 + p) p     ζ(k + 1) k+1 Y 1 k+ 21 +ε = . +O x x 1− k (k + 1)ζ(2) p (1 + p) p  B(x) = P (SP (n)) , |^ Abel Úúª, Œ k n≤x X nl (SP (n))k n≤x l Z x B(t)tl−1 dt 1     1 ζ(k + 1) k+l+1 Y k+l+ 21 +ε +O x x 1− k = (k + 1)ζ(2) p (1 + p) p  Z x Z x  1 lζ(k + 1) Y k+l k+l− 12 +ε 1− k t dt + O t − dt (k + 1)ζ(2) p p (1 + p) 1 1    Y 1 1 ζ(k + 1) k+l+1 + O xk+l+ 2 +ε , x 1− k = (k + l + 1)ζ(2) p (1 + p) p = x B(x) − 1 − l X (SP (n))k n≤x −l nl = x B(x) − 1 + l Z x B(t)t−l−1 dt 1 119 'uSmarandache¯KïÄ#?     1 ζ(k + 1) k−l+1 Y k−l+ 21 +ε +O x x 1− k = (k + 1)ζ(2) p (1 + p) p   Z x Z x 1 lζ(k + 1) Y k−l k−l− 12 +ε dt 1− k t dt + O t + (k + 1)ζ(2) p p (1 + p) 1 1    Y ζ(k + 1) 1 k−l+ 21 +ε k−l+1 = . +O x x 1− k (k − l + 1)ζ(2) p (1 + p) p ½ny. Šâ½n,  l = 0, k = k1 , =ŒíØ 7.3.1;  k = 1, 2, 3, Ä  ζ(2) = π /6, ζ(4) = π /90, =ŒyíØ 7.3.2. Œ±wÑ, T½n´ é©z [63] í2. ′ 2 4 'u Smarandache {ü¼ê 7.4 F. Smarandache : Ç3©z [1] ¥1 42 ‡¯K½Â Smarandache {ü¼êXe ½Â 7.3.  n Ǒê, Smarandache {ü¼ê S (n) ½ÂǑ: ÷ v p |m! ê m ∈ N, =µ p n Sp (n) = min {m : pn |m!, m ∈ N} . ©z [68] ½Â Smarandache {ü¼ê\{aq¼êXe: ½Â 7.4.  S (n) = min {m : p ≤ m!!, m ∈ N} (n ∈ (1, ∞)) Ú S (n) = max {m : m!! ≤ p , m ∈ N} (n ∈ (1, ∞)), K¡ S (n) Ú S (n) Ǒ Smarandache {ü¼ê\{aq. w,, e (m − 2)!! < p ≤ m!!, S (n) = m, Ù¥ m > 2. 'u S (n) 5Ÿ, NõÆöÑ?1 ïÄ, 넩z [69-72]. X©z [72] ïÄ d(S (n)) þŠ5Ÿ, Ñ ìCúª: n p ∗ P ∗ p n n p p p X n≤x 120 d(S p (n)) = 2x(ln x − 2 ln ln x) + O (x ln p) . p 1ÔÙ ¼êƒ'¯K !̇ïÄ σ (S (n)) ìC5Ÿ, Ù¥ σ (n) = P d ´ØêÚ¼ ê, ¿  ü‡Ǒ°(ìCúª. ½n 7.5.  p Ǒ˜‡‰½ƒê , é?¿¢ê x ≥ 1, k      π x ln p 2x ln p x   ln , XJα = 1, +O   ln x  3 ln x ln x X α Smarandache p α α d|n [73] 2 2 2 2 σα (S p (n)) = n≤x ζ(α + 1) 2   α+1    2 α+1 α+1 α  2x ln p x ln p ln α+1 ln x ln x   xα+1 , +O lnα+1 x α 6= 1.  XJ ½n 7.6.  p Ǒ˜‡‰½ƒê , é?¿¢ê x ≥ 1, k      π x ln p 2x ln p x   ln , XJα = 1, +O   ln x  3 ln x ln x X [73] 2 2 2 2 ∗ σα (S p (n)) n≤x = 2 ζ(α + 1) 2   α+1    α+1 α+1 α x ln p ln α+1 ln x  2x ln p ln x   xα+1 +O lnα+1 x α 6= 1. XJ  Ún 7.4.1. é?¿¢ê x ≥ 1, k X σ1 (n) = n≤x π2 2 x + O (x ln x) . 12 Ún 7.4.2. é?¿¢ê x ≥ 1 Ú α > 0, α 6= 1, k X σα (n) = n≤x
ζ(α + 1) α+1 x + O xβ , α+1 Ù¥ β = max{1, α}. Ún 7.4.1 ÚÚn 7.4.2 y²„©z [8].   ln(m − 2)!! ln m!! ½ny²: d S (n) ½Â,  n ∈ ln p , ln p , k S (n) = m. e (m − 2)!! < p ≤ m!!, K m − 2xlnlnx p ≪ ln ln x. dÚ n 7.4.1 Ú Abel ª, k p x p X σ1 (S p (n)) n≤x 121 , 'uSmarandache¯KïÄ#? = X X = X m≤ 2xlnlnx p = X m≤ 2xlnlnx p = X m≤ 2xlnlnx p 1 = ln p   ln m  σ1 (m) + O  ln p 2x ln p ln x  X <m< 2xlnlnx p +ln ln x ln x  σ1 (m) ln p  ln m σ1 (m) + O (x ln ln x) ln p   ln m   X σ1 (m) + O (x ln ln x) σ1 (m) + O  ln p 2x ln p  m≤ X ln mσ1 (m) + O m≤ 2xlnlnx p  σ1 (m) 2x ln p <m< 2xlnlnx p +ln ln x ln(m−2)!! <n≤ lnlnm!! ln x ln p p m≤ 2xlnlnx p ln(m−2)!! <n≤ lnlnm!! ln p p  X X σ1 (m) +  2x ln p 1  = ln ln p ln x  X m≤ 2xlnlnx p ln x  x2 ln2 x  σ1 (m) − Z  2x ln p ln x 1X  σ1 (t)dt + O t 1 m≤t  x2 ln2 x     Z 2x ln p  2 ln x 1 2x ln p π 2 4x2 ln2 p 1 π 2 1 − ln t + O (t ln t) dt = ln p ln x 12 ln2 x ln p 1 t 12  2  x +O ln2 x   2   π 2 x2 ln p x 2x ln p = +O ln . 2 3 ln x ln x ln2 x XJ α 6= 1, dÚn 7.4.2 Ú Abel ª, k X σα (S p (n)) n≤x = X X m≤ 2xlnlnx p ln(m−2)!! <n≤ lnlnm!! ln p p = X m≤ 2xlnlnx p = X m≤ 2xlnlnx p 122  ln x  X 2x ln p <m< 2xlnlnx p +ln ln x ln(m−2)!! <n≤ lnlnm!! ln x ln p p ln m  σα (m) + O  ln p 2x ln p  X σα (m) + X <m< 2xlnlnx p +ln ln x    ln x  σα (m) ln p ln m  X  σα (m) + O  σα (m) + O (x ln ln x) ln p 2x ln p m≤ ln x σα (m) 1ÔÙ 1 = ln p X ln mσα (m) + O m≤ 2xlnlnx p   2x ln p 1  ln ln p ln x = Smarandache  X m≤ 2xlnlnx p  ¼êƒ'¯K xα+1 lnα+1 x σα (m) −  Z 2x ln p ln x 1  1X  σα (t)dt t m≤t  xα+1 +O lnα+1 x   2x ln p ζ(α + 1) 2α+1 xα+1 lnα+1 p 1 ln = ln p ln x α+1 lnα+1 x    α+1  2x ln p Z ln x 1 ζ(α + 1) 2 1 x α − t + O (t ) dt + O ln p 1 t α+1 lnα+1 x     xα+1 ζ(α + 1) 2α+1xα+1 lnα+1 p 2x ln p +O = ln . α+1 ln x lnα+1 x lnα+1 x  ùÒ¤ ½n 7.5 y². ^Ӑ{Œ±y²½n 7.6. Smarandache k 7.5 gÖê¼ê ½Â 7.5.  k ≥ 2 Ǒê, éu?¿ê n ≥ 2, e A (n) ´ ÷v A (n) × n Ǒ k gê, ¡ A (n) Ǒ n  k g Öê¼ê, Ǒ¡ A (n) Ǒ n  k gÖê. ~X, A (1) = 1, A (2) = 2, A (3) = 3, A (4) = 1, A (5) = 5, A (6) = 6, A (7) = 7, A (8) = 2, · · · , = A (2) = 2 , A (3) = 3 , A (2 ) = 1, · · · . ½Â 7.6. é?¿ê k ≥ 2, a (n) ´÷v a (n) + n Ǒ k g ê, ¡ a (n) Ǒ n  k gŒ\Öê¼ê, Ǒ¡ a (n) Ǒ n  k gŒ\Öê. ½Â 7.7. é?¿ê n, f (n) = min{r : 0 ≤ r = n − m , m ∈ N}, ¡ f (n) Ǒ n  k g~{Ö¼ê, = f (n) ´šKê, n − f (n) ´˜ ‡ k g?ۚKŽê¼ê h(n) ê. ½¡ f (n) Ǒ n  k g~{Öê. k k k k 2 k 2 k−1 2 2 k−1 k k 2 k 2 k 2 2 k k k k k k k k k 123 'uSmarandache¯KïÄ#? ½Â 7.8. éê n, eÙIO©)ªǑ n = p p · · · p , ½Â ( α + α + · · · + α , n > 1, Ω(n) = n = 1. 0, é Smarandache  k gÖê¯K, kéõÆö®²‰LïÄ¿¼ ˜ k(J , X©z [74] éŒ\ k gŒ\Öê a (n) ‰ Ñ ìCúª: é?¿¢ê x ≥ 3, α1 α2 1 2 1 2 αk k k [74−76] X k ak (n) = n≤x   1 2 k2 x2− k + O x2− k . 4k − 2   1 1 1 d(ak (n)) = (1 − )x ln x + (2γ + ln k − 2 + ) + O x1− k ln x , k k n≤x X Ù¥ d(n) Ǒ Dirichlet Øê¼ê, γ Ǒî.~ê. 3©z [74] Ä:þ, !$^Ú)ې{ïÄ n − f (n)  þŠ9Ú5Ÿ, ¼ ˜ kìCúª („©z [78]), * F. Smarandache Ç3 [1] ˜Ö¥¤9¯KïÄóŠ. ½n 7.7. éu?Û¢ê x > 1, ke¡ìCúª k  x  , Ω(n − fk (n)) = kx ln ln x + k(A − ln k)x + A2 x + O ln x n≤x X Ù¥, A = γ + X(ln(1 − p1 ) + p −1 1 ), P L«ƒêƒÚ, γ ´î.~ê. ½n p p é u ? Û ¢ ê x ≥ 3 Ú  ê k ≥ 2, é ?  α ≤ 1 ž, ?ê´uÑ;  α > 1 ž, ?ê 7.8. 1 , α n=1 (n − fk (n)) , ê ´Âñ ∞ P ∞ X 1 = Ck1 ζ(kα − k + 1) + Ck2 ζ(kα − k + 2) + · · · α (n − f (n)) k n=1 +Ck2 ζ(kα − 2) + Ck1 ζ(kα − 1) + ζ(kα), Ù¥, ζ(s) ´ Riemann zeta- ¼ê, AO k = α = 2, 9 k = α = 3, Œ íÑ X ∞ 1 = 2ζ(3) + ζ(4), (n − f2 (n))2 n=1 124 1ÔÙ ¼êƒ'¯K Smarandache ∞ X 1 = 3ζ(7) + 3ζ(8) + ζ(9)2ζ(3) + ζ(4). 3 (n − f (n)) 3 n=1 ‡¤½ny²I‡±eÚn: Ún 7.5.1. é?¿¢ê x ≥ 1, kìCúª X Ω(n) = x ln ln x + Ax + O n≤x  x  , ln x Ù¥, A = γ + X(ln(1 − p1 ) + p −1 1 ), P L«ƒêƒÚ, γ ´î.~ê. p p y²: „©z [77]. ½ny²: Äk5y²½n 7.7. éu?¿ê x ≥ 2, 3 ê M ,  M ≤ x < (M + 1) , Œ±íä, - M = [x ], ¿5¿  x − M = O(1), é?¿ƒê p 9Ù­ê α, 5¿ Ω(p ) = αp 9 k k 1/k 1/k α k (x + 1) = X n≤x Ω(n − fk (n)) M−1 X X M−1 X X t=1 tk ≤n≤(t+1)k = t=1 tk ≤n≤(t+1)k = Cki xk−i, i=0 Œ±íä = k X M−1 X t=1  Ω(n − fk (n)) + O   Ω((t + 1)k ) + O  X M k ≤n<x X  Ω(n − fk (n)) M k ≤n<(M+1)k  Ω((M + 1)k ) k(Ck1 tk−1 + Ck2 tk−2 + Ck3 tk−3 + · · · + Ck1 t1 + 1)Ω(t + 1)   +O x(k−1)/k + ε = k2 M−1 X t=1   (t + 1)k−1Ω(t + 1) + O x(k−1)/k + ε 125 'uSmarandache¯KïÄ#? M   X (t)k−1 Ω(t) + O x(k−1)/k + ε , = k 2 t=1 Ω(n) ≪ nε , - A(x) = P Ω(n), ^ Abel Úúª´ M X (t)k−1 Ω(t) t=1 = = = M ′ A(M ) − A(t)((t)k−1 ) dt + O (1) 2    M k−1 M ln ln M + AM + O M ln M   Z M t − t ln ln t + At + O (k − 1)tk−2 dt + O (1) ln t 2  k  M M k ln ln M + AM k + O ln M Z M − ((k − 1)tk−1 ln ln t + A(k − 1)tk−1 )dt 2  k  M k−1 k k k k (M ln ln M + AM ) + O M ln ln M + AM − k ln M  k  1 k A M , M ln ln M + M k + O k K ln M = M = Z k−1 du 0 ≤ x − M k < (M + 1)k − M k = Ck1 M k−1 + Ck2 M k−2 + Ck3 M k−3 + · · · + Ck1 M 1 + 1 ≤ x(k−1)/k , ln k + ln ln M ≤ ln ln x < ln k + ln ln(M + 1) <
ln k + ln ln M + O x−1/k , l k M  x  X 1 , (t)k−1 Ω(t) = x ln ln M + (A − ln k)xk + O k ln x t=1 X n≤x Ω(n − fk (n)) = x ln ln x + k(A − ln k)x + O ùÒ¤½n 7.7 y². 126  x  . ln x 1ÔÙ Smarandache ¼êƒ'¯K ey½n 7.8. é?¿ê n ≥ 1, 3ê m,  m ≤ n < (m + 1) , Œ±íä, ÷v n − f (n) = m ª n
k ((m + 1) − m ), k k ∞ X k k k k ∞ X 1 (m + 1)k − mk = (n − fk (n))α mkα n=1 m=1 ∞ X Ck1 mk−1 + Ck2 mk−2 + + · · · + Ck2 m2 + Ck1 m1 + 1 = mkα m=1 d‘?ê"ñ{é?Û¢ê α ≤ 1, ?ê´uÑ, XJ α > 1, ?ê´Âñ, ÙÚǑ Ck1 ζ(kα−k+1)+Ck2 ζ(kα−k+2)+· · ·+Ck2 ζ(kα−2)+Ck1 ζ(kα−1)+ζ(kα). ùÒ¤½n 7.8 y². 3½n 7.8 ¥ k = α = 2 9 k = α = 3 =íØ. 7.6 ˜‡¹ Gauss ¼ê§9Ù¢ê) QJÑXe¯K: ϐ§ F. Smarandache 1. ¯K xy − [x] = y (7-8) ¤k¢ê), Ù¥ [x] ´Ø‡L x Œê. éd¯KR, Smarandache ¿™)û. ~X±eœ¹ÿ™?Ø: (a) y ∈ ; Q m Q ∈ ; (b) y = n Z 5: Smarandache Ñe y ´Œu 1 Ûê, Kk 1 x = (y + 1) y . (7-9) éudœ¹, ª (7-9) ØɁ›. Ïe y > 0, Kk y + 1 < (1 + 1)y = 2y . 127 'uSmarandache¯KïÄ#? òª (7-9) “\§ (7-8), u´k y 1 xy − [x] = (y + 1) y − ⌊(y + 1) y ⌋ = y + 1 − 1 = y. (7-10) ª (7-10) L² 0 < y ∈ R ž, § (7-8) )Ǒ x = (y + 1) .  y < 0 ž, ÏǑ‡ÄEê, ¤±Ï§ (7-8) )¬'(J, ùk–·‚?˜Ú?Ø. ¯K 2. ϐ§ x − [x] = y ¤k¢ê). ¯K 3. ϐ§ x − [x] = x ¤k¢ê). ¯K 4. ϐ§ x[y] − [x]y = |x − y| ¤k¢ê). ¯K 5. ϐ§ x − y = |x − y| ¤k¢ê). ¯K 4 ®ÿt“)û, „©z [79]. C, <a [80] |^ {鐧 x − [x] = x Œ)5?1 ïÄ, ӞA^ Mathematic 5.0 ^‡Œ±yT§3z‡«m [n, n + 1] (n ∈ N) þÑ3¢ê). du x, y Cz5š~Œ, éJ‰Ñ§¤k)äN/ª, ¤±¨= Ä x, y > 0 žœ¹. éu x, y < 0 žœ¹, Œ±|^é¡Ñ. 3 y ≥ 1 ž, §þk¢ê). éu y éêž, Œ±½ y Š, r§=zǑ'u x §, , )Ñ x. äN`Ò´y² ±e: 7.6.1. ng§ s x r− px − q = 0 (p, q > 0) ¢ê)Ǒ x = s Ún r 1 y y y y y y [y] [x] y 3 q + 2 3 p q ( )2 − ( )3 + 2 3 3 q − 2 p q ( )2 − ( )3 . 2 3 ½n 7.9. é?¿ê N, M 9‰½ y ∈ Q \{0 < y ≤ 2},  § x − [x] = x 3«m [N, M ] þk k M − N + O(1) ‡¢ê). y²: - f (x) = x −[x] −x.  y > 2 ž, 3«m [n, n+1) (n ∈ Z ) S, f (n) = n − n − n < 0, q lim f (x) = (n + 1) − n − n − 1 = y 1 n (1 + ) − n − n − 1, ÏǑ (1 + ) > 1 + ,  lim f (x) = n n + y y y y y + y y y x→(n+1)− y 1 y n y y x→(n+1)− 1 ny (1 + )y − ny − n − 1 > ny + yny−1 − ny − n − 1 = yny−1 − n − 1 > 0. n 0 < δ < 1, n < x < n + δ , f (x) > 0 . =3 128  ž ¤á 1ÔÙ Smarandache ¼êƒ'¯K Ӟ- f (θ) = (n + θ) − n − y (0 ≤ θ < 1),  f (θ) = y(n + θ) > 0. ¤±¼ê f (x) 3 [n, n + δ] SëY4O. |^":3½n Œ, 3z‡«m [n, n + δ] þ, f (x) = 0 k k˜‡). ǑÒ´`,  x ∈ [N, M ], N, M ∈ N ž, § x −[x] = x k k M −N +O(1) ‡¢ê). Ïd§ x − [x] = x k ¡õ¢ê). y ′ y y−1 + y y y y íØ 7.6.1. § x y − [x]y = x k¢ê) ( x ∈ [0, 1), y = 1. y²:  y = 1 ž, § x − [x] = x CǑ (x − [x] = x, = [x] = 0. ¤± x ∈ [0, 1). u´§ x − [x] = x k¢ê) yx =∈ [0,1. 1), y y íØ 9 y y § 7.6.2. xy − [x]y  √ 1 + 1 + 4n2  x= (n ∈ N+ ), 2  y = 2. = x k¢ê) ( x=0 y=2 y²:  y = 2 ž, § x − [x] = x =Ǒ˜g§ x − [x] − x = 0. e x = 0 ž( , § x − [x] − x = 0 w,¤á, ¤± § x − [x] = x k¢ê) xy == 2.0, e x√∈ [n, n + 1), x − [x] = x §CǑ x − x − n = 0. )ƒ, y 2 2 y y 2 2 y y y 2 2 1 + 4n2 (n ∈ N+ ). 2 √ √ 1 + 1 + 4n2 , 2n < 1 + 4n2 < 2n + 1. √ 2 √ 1 + 1 + 4n2 1 + 1 + 4n2 − 2n > 0, n + 1 − n = 2 √ 2 √ 1 + 1 + 4n2 2n + 2 − 1 − 1 + 4n2 > 0 (n ∈ N+ ), n≤x= 2 2 √ 1 + 1 + 4n2 n + 1, (n ∈ N+ ) x = x2 − x − n2 = 2  √ 1 + 1 + 4n2  (n ∈ N+ ) x= xy − [x]y = x . 2  y=2 x= 1+ w,  = du ¤± ÷v§ ´§ − = < 0, ¢ê) 129 'uSmarandache¯KïÄ#? § íØ 7.6.3. xy − [x]y = x k¢ê) ( x=0 y=3   r 9 x = 3 n3 2 y = 3. + q n6 4 − 1 27 + r 3 n3 2 − q n6 4 − 1 27 (n ∈ N+ ), y²:  y = 3 ž, § ( x − [x] = x =Ǒ x − x − [x] § x − [x] = x k¢ê) xy == 3.0, e x ∈ [n, n + 1) (n ∈ N ), § x − [x]r = x qCǑ x x = 0, |^ÚnŒ x = A + B, Ù¥: A = + − r y y y 3 y y 3 n3 2 − = 0, 3 − n3 − y + 3  3 q w, n6 4 − n3 2 n6 4 1 27 , B = 1 27 . s r s r 3 3 n n6 1 1 A+B = + − + − − 2 4 27 2 4 27 v ! ! r r u 6 6 3 u n3 n n 1 1 n 3 + + − − − > t 2 4 27 2 4 27 3 n3 n6 = n, ÏǑ (A + B) = A + B ¤± A + B ≥ n. ÏǑ 3 3 s 3 + 3A2 B + 3AB 2 = n3 + (A + B), A > 0, B > 0, 130 r s r 3 3 n n6 1 1 + − + − − 2 4 27 2 4 27 v ! r u u n3 1 n6 3 + − < 2t 2 4 27 " 1/3  1/6 # 3 6 n n 1 < 2 + + 2 4 27 3 n3 n6 1ÔÙ < 2  ¼êƒ'¯K Smarandache  n √ + n < n + 2n = 3n, 3 2  (A + B) =rn + (Aq+ B) < n +r3n < q(n + 1) , = A + B < n + 1. l x = + − + − − ∈ [n, n + 1), = 3 3 3    ´§ x y n 7.7 x= n3 2 r 3 3 n3 2 n6 4 + y = 3. − [x]y = x q 3 1 27 n6 4 − 3 1 27 + n3 2 r 3 n3 2 n6 4 − q n6 4 1 27 − 1 27 (n ∈ N+ ), ¢ê). ?›¥š"êi겐ڼêþŠ Ç3©z [1] ¥JÑ1 22 ‡¯K´ “ïě ?›¥êiƒÚê5Ÿ”. ©z [81-83] òù˜¯K˜„z, ̇ ïÄ n ?›¥êiƒÚ¼ê!êi²Ú¼êþŠ. ÊwÚ1 ¯ [84] 3dÄ:þ, ÏLínØy‰Ñ n ?›¥š"êiê²Ú ¼ê a(m, n) þŠ, = A(m, n) °(OŽúª. Ǒ QãB, Ú\ Xe½Â: ½Â 7.9.  n (n ≥ 2) Ǒ˜‰½ê, é?˜ê m, b ½ m 3 n ?›¥L«Ǒ m = a n + a n + · · · + a n , Ù¥11 ≤ a1 ≤ + + n − 1, i = 1, 2, · · · , s, k > k > · · · > k ≥ 0, K¡ a(m, n) = a a P 1 ··· + Ǒ n ?›¥š"êi겐ڼê, A(N, n) = a(m, n) a Ǒ¼ê a(m, n) þŠ. Ǒ {zúª, P ϕ ( n1 ) = X i1 . F. Smarandache 1 1 k1 2 2 k2 s ks s i 2 1 2 s 2 2 m<N n−1 r r ½n 7.10.  N = a n Ù¥ + a2 nk2 + · · · + as nks , 1 ≤ ai < n, i = 1, 2, · · · , s, k1 > k2 > · · · > ks ≥ 0,    i−1 s  X X ki ai ϕ2 ( n1 ) 1  ki 1  + ϕ2 ( ) + ai n . A(N, n) =  n ai a2  j=1 j i=1 1 k1 i=1 Kk 131 'uSmarandache¯KïÄ#? AO n = 2 ž, kXeíØ: íØ 7.7.1.  N = 2 +2 +· · ·+2 , Ù¥ k > k K  X k1 k2 s A(N, 2) = i=1 ks 1 2 > · · · > ks ≥ 0, ki + (i − 1) 2ki . 2 Ǒ ¤½ny², I‡Ú\e¡ü‡Ún: Ún 7.7.1. 1 y²:  A(nk , n) = knk−1 ϕ2 ( ). (7-11) n X k = 1 , =A(n, n) = a(m, n) = a(1, n) + ž †> m<N 1 1 1 1 = ϕ ( )= a(2, n) + · · · + a(n − 1, n) = 2 + 2 + · · · + 2 1 2 (n − 1)2 n . ·K¤á b k = p ž·K¤á, = 1 A(np , n) = pnp−1 ϕ2 ( ). n o k = p + 1 ž, A(np+1 , n) = X m>,  (7-12) a(m, n) m<np+1 = X a(m, n) + m<np = X np ≤m<2np a(m, n) + m<np + X X 0≤m<np X 0≤m<np a(m, n) + · · · + X (n−1)np ≤m<np+1 a(m + np , n) + · · · a(m + (n − 1)np , n)  X  1 = a(m, n) + a(m, n) + 2 + · · · 1 p p m<n 0≤m<n  X  1 a(m, n) + + (n − 1)2 0≤m<np   X 1 1 1 + 2 + ··· + np = n a(m, n) + 2 2 1 2 (n − 1) m<np X 132 a(m, n) 1ÔÙ Smarandache ¼êƒ'¯K 1 = nA(np , n) + ϕ2 ( )np . n (7-13) d (7-12) ªÚ (7-13) ª,  1 1 1 A(np+1 , n) = npnp−1 ϕ2 ( ) + ϕ2 ( )np = (p + 1)ϕ2 ( )np . n n n ¤±,  k = p + 1 ž·K¤á. u´¤ Ún 7.7.1 y². Ún 7.7.2. 1 1 A(bnk , n) = bknk−1 ϕ2 ( ) + ϕ2 ( )nk , n b Ù¥ b Ǒg,ê. y²: A(bnk , n) = X a(m, n) m<bnk = X a(m, n) + m<nk = X nk ≤m<2nk a(m, n) + m<nk + X X 0≤m<nk X 0≤m<nk a(m, n) + · · · + X a(m, n) (b−1)nk ≤m<bnk a(m + nk , n) + · · · a(m + (b − 1)nk , n)  X  1 = a(m, n) + a(m, n) + 2 + · · · 1 m<nk 0≤m<nk   X 1 + a(m, n) + (b − 1)2 0≤m<nk   X 1 1 1 + 2 + ··· + nk = b a(m, n) + 2 2 1 2 (b − 1) k X m<n 1 = bA(nk , n) + ϕ2 ( )nk . b (7-14) d (7-11) ªÚ (7-14) ª,  1 1 A(bnk , n) = bknk−1 ϕ2 ( ) + ϕ2 ( )nk . n b (7-15) u´¤ Ún 7.7.2 y². 133 'uSmarandache¯KïÄ#? ½ny²: A(N, n) = X a(m, n) m<N X = m<a1 nk1 + X a(m, n) + X a1 nk1 ≤m<a1 nk1 +a2 nk2 a(m, n) + · · · a(m, n) N−as nks ≤m<N   1 = a(m, n) + a(m, n) + 2 + · · · a1 k k 1 2 m<a1 n 0≤m<a2 n ! s−1 X X 1 a(m, n) + + a2 i=1 i 0≤m<as nks   s s i−1 X X X 1  = A(ai nki , n) + ai nki . 2 a i=1 i=1 j=1 j X X (7-16) d (7-15) ªÚ (7-16) ª,     X i−1 s s  X X 1 1 1  ai nki ai ki nki −1 ϕ2 ( ) + ϕ2 ( )nki + A(N, n) = 2 n a a i j=1 j i=1 i=1    s i−1 X X ai ki ϕ2 ( n1 ) 1   ki 1   = + ϕ2 ( ) + ai n . n ai a2 i=1 j=1 j u´¤ ½ny². 134 ë©z ë©z [1] F. Smarandache, Only Problems, Not Solutions, Chicago: Xiquan Publishing House, 1993. [2] Liu Yaming, On the solutions of an equation involving the Smarandache function, Scientia Magna, 2(2006), No. 1, 76-79. , Smarandache , , 49(2006), No. 5, 1009-1012. [3]
[4] Le Mohua, A lower bound for S 2p−1 (2p − 1) , Smarandache Notions Journal, 12(2001), No. 1-2-3, 217-218. [5] , Smarandache , , 22(2009), No. 1, 133-134. [6] , Smarandache , , 24(2008), No. 4, 706-708. [7] Wang Jinrui, On the Smarandache function and the Fermat numbers, Scientia Magna, 4(2008), No. 2, 25-28. [8] Tom. Apostol, Introduction to Analytic Number Theory, New York: SpringerVerlag, 1976. [9] , , , , 2007. , Smarandache , , 26(2010), No. 3, [10] 413-416. Mó¸ €ïw 'u €ïw 'u ¼êŠ©Ù êÆÆ ¼ê˜‡e.O —„pÄ:‰ÆÆ ¼ê˜‡#e.O X{êƆA^êÆ Ü©+ êØ ñܓ‰ŒÆч ÜS §X¶ ¼ê˜‡e.O X{êƆA^êÆ [11] Á¯, 'u Smarandache ¼ê†¤êê, ÜŒÆÆ (g,‰Æ‡), 40(2010), No. 4, 583-585. [12] J. Sándor and F. Luca, On S(n! + 1), where S is the Smarandache function, Octogon Math. Mag., 16(2008), No. 2, 1024-1026. [13] C. L. Stewart, On the greatest and least prime factors of n! + 1, II, Publ. Math. Debrecen, 65(2004), No. 3-4, 461-480. [14] M. Murthy and S. Wong, The abc-conjecture and prime divisors of the Lucas and Lehmer sequences, Peters, K., Number theory theory for the millenium III, Natick, MA, 2002, 43-54. [15] J. Oesterlé, Nouvelles approaches du Théoréme Fermat, Sem. Bourbaki, 1987/1988, No. 694, 1-21. [16] D. W. Masser, Open problems, Analytic number theory, London: Imperial College, 1995, 27-35. [17] R. K. Guy, Unsolved problems in number theory, third edition, New York: Springer Verlag, 2004. [18] P. Erdös and C. L. Stewart, On the greatest and least prime factors of n! + 1, J. London Math. Soc., 13(1976), No. 4, 513-519. [19] J. Sándor, On certain new inequalities and limits for the Smarandache function, Smarandache Notions Journal, 9(1998), No. 1-2, 63-69. [20] , , : , 1981. [21] Chen Jianbin, Value distribution of the F. Smarandache LCM function, Scientia ç5Ú |ÜPê{9ÙA^ ® ‰Æч 135 'uSmarandache¯KïÄ#? Magna, 3(2007), No. 2, 15-18. [22] Le Maohua, Two function equations, Smarandache Notions Journal, 14(2004), No. 1-3, 180-182. [23] , Smarandache LCM , , 24(2008), No. 1, 71-74. [24] Lu Zhongtian, On the F. Smarandache LCM function and its mean value, Scientia Magna, 3(2007), No. 1, 22-25. [25] Ge Jian, Mean value of the F. Smarandache LCM function, Scientia Magna, 3(2007), No. 2, 109-112. [26] , Smarandache LCM , ( ), 39(2010), No. 3, 229-231. [27] , , , : , 1988. ë 'u ¼ê˜aþ þŠ X{êƆA^êÆ A¡_ ¼ê†Ùéó¼ê·ÜþŠ S„“‰ŒÆÆ g ,‰ÆÇ©‡ «É «J ƒê½ny² þ° þ°‰ÆEâч [28] A¡_, Smarandache LCM éó¼ê†ƒÏf¼êþŠ, —„pÄ: ‰ÆÆ, 23(2010), No. 3, 323-325. [29] I, ˜‡¹# Smarandache ¼ê§, êÆ¢‚†£, 40(2010), No. 11, 206-210. [30] ²^, 'u Smarandache ¼ê9 Smarandache LCM ¼ê·ÜþŠ, ÜŒÆ Æ (g,‰Æ‡), 40(2010), No. 5, 772-773. [31] I, ˜‡¹Žê¼ê S(d) Ú SL(d) §, ÜŒÆÆ (g,‰Æ‡), 40(2010), No. 3, 382-384. [32] http://www.fs.gallup.unm.edu/mathematics.htm. [33] http://www.gallup.unm.edu/∼smarandache/Smarandacheials.htm. [34] , Smarandache Dirichlet , ( ), 38(2010), No. 5, 14-17. [35] C. H. Zhong, A sum related to a class arithmetical functions, Utilitas Math., ï² ˜a¹ ‰Æ‡ Ú¼ê ?ê ñܓ‰ŒÆÆ g, 44(1993), 231-242. [36] H. N. Shapiro, Introduction to the theory of numbers, John Wiley and Sons, 1983. [37] M. L. Perez, Florent Smarandache Definitions, Solved and Unsolved Problems, Conjectures and Theorems in Number theory and Geometry, Chicago: Xiquan Publishing House, 2000. [38] Liu Hongyan and Zhang Wenpeng, On the simple numbers and its mean value properties, Smarandache Notions Journal, 14(2004), 171-175. [39] Zhu Weiyi, On the divisor product sequences, Smarandache Notions Journal, 14(2004), 144-146. [40] Kenichiro Kashihara, Comments and Topics on Smarandache Notions and Problems, USA: Erhus University Press, 1996. [41] , SSSP (n) SISP (n) , , 25(2009), No. 3, 431-434. ƒ 'u Ú þŠ X{êƆA^êÆ [42] o®Ú, Smarandache ²ê SP (n) Ú IP (n) þŠ , X{êƆA^êÆ, 26(2010), No. 1, 69-71. [43] Lu Xiaoping, On the F. Smarandache 3n-digital sequence, Research on Number Theory and Smarandache Notions, edited by Zhang Wenpeng, USA: Hexis, 2009, 5-7. [44] Wu Nan, On the Smarandache 3n-digital sequence and the Wenpeng Zhang’s 136 ë©z conjecture, Scientia Magna, 4(2008), No. 4, 120-122. [45] Yang Ming, The conjecture of Wenpeng Zhang with respect to the Smarandache 3n-digital sequence, Research on Number Theory and Smarandache Notions, edited by Zhang Wenpeng, USA: Hexis, 2010, 57-61. [46] , Smarandache , , 27(2010), No. 4, 446-448. [47] Wu Xin, An equation involving the Pseudo-Smarandache function and F. Smarandache LCM function, Scientia Magna, 5(2009), No. 4, 41-46. [48] , , Smarandache Smarandache , ( ), 33(2010), No. 2, 200-202. [49] Murthy A., Smarandache reciprocal function and an elementary inequality, Smarandache Notions Journal, 11(2000), 312-315. oçï ˜‡¹ü‡ ¼ê§9Ùê) ç9ŒÆg,‰Æ Æ o ƒ‘| ˜a¹ ê) oA“‰ŒÆÆ g,‰Æ‡ ¼ê†– ¼ê§9Ù [50] J. Sándor, On certain arithmetic function, Smarandache Notions Journal, 12(2001), 260-261. [51] J. Sándor, On a dual of the Pseudo Smarandache function, Smarandache Notions Journal, 13(2002), 18-23. [52] , F. Smarandache , , 38(2008), No. 2, 173-175. [53] Wang Yongxing, Some identities involving the Smarandache ceil function, Scientia Magna, 2(2006), No. 1, 45-49. [54] Lu Yaming, On a dual function of the Smarandache ceil function, Research on Smarandache problems in number theory, edited by Zhang Wenpeng, USA: Hexis, 2005, 55-57. [55] Ding Liping, On the mean value of the Smarandache ceil function, Scientia Magna, 2(2005), No. 1, 74-77. Ü©+ 'u ¼êü‡¯K ÜŒÆÆ [56] Zhang Wenpeng, On an equation of Smarandache and its integer solutions, Smarandache Notions, 13(2002), 176-178. [57] , , : , 1983. [58] , Smarandache , ( ), 37(2007), No. 2, 197-198. , Smarandache , ( ), 40(2010), [59] No. 1, 9-10. [60] , , ( ), 38(2010), No. 2, 12-14. [61] Russo F., An introduction to the Smarandache square complementary function, Smarandache Notions Journal, 13(2002), No. 1-2-3, 160-172. [62] , SSC(n) , , 25(2010), No. 3, 436-437. ‡ E ŒÆêÆX êÆ©Û ® p˜Ñ‡ 4ÿV ˜‡¹ ¼ê§ ÜŒÆÆ g,‰Æ‡ H¡ÿ 'u ¯K˜‡í2 ÜŒÆÆ g,‰Æ‡ H¡ÿ ˜‡Žâ¼ê†ŒƒÏf¼ê·ÜþŠ ñܓ‰ŒÆÆ g,‰Æ RŸ 'u²Ö¼ê ü‡¯K Hà’ŒÆÆ [63] Mó¸, Smarandache ˜¼êþŠ, êÆÆ, 49(2006), No. 1, 77-80. [64] Zhou Huanqin, An infinite series involving the Smarandache power function SP (n), Scientia Magna, 3(2006), No. 2, 109-112. [65] Tian C. and Li X., On the Smarandache power function and Euler totient function, Scientia Magna, 4(2008), No. 1, 35-38. 137 'uSmarandache¯KïÄ#? «É «J )ÛêØÄ: ® ‰Æч ?+  "Öw 'u ˜¼ê5P ‰E [66] , , , : , 1999. [67] , , , Smarandache , , 28(2010), No. 17, 50-53. [68] J. Sándor, On additive analogue of certain arithmetic function, Smarandache Notions Journal, 14(2004), 128-132. [69] Le Maohua, Some problems concerning the Smarandache square complementary function, Smarandache Notions Journal, 14(2004), No. 1-3, 220-222. [70] Zhu Minhui, The additive analogue of Smarandache simple function, Research on Smarandache problems in number theory, edited by Zhang Wenpeng, USA: Hexis, 2004, 39-40. [71] , , , 19(2006), No. 1, 5-6. [72] , F. Smarandache , , 27(2009), No. 3, 337-338. F[r êؼê9ِ§ —„pÄ:‰ÆÆ Á¯ 'u ¼ê˜‡¯K Ü‰Æ [73] Á¯, 'u F. Smarandache {ü¼ê† Dirichlet ØêÚ¼ê·ÜþŠ, S„ “‰ŒÆÆ (g,‰ÆÇ©‡), 39(2010), No. 5, 441-443. [74] Xu Zhefeng, On additive k-power complements, Research on Smarandache problems in number theory, edited by Zhang Wenpeng, USA: Hexis, 2004, 11-14. [75] , , , k , , 23(2007), No. 3, 347-350. [76] Liu Hongyan, Liu Yuanbing, A note on the 29th Smarandache problem, Smarandache Notions Journal, 14(2004), No. 1-3, 156-158. [77] Hardy G. H., Ranlnanujan S., The normal number of prime factors of a number n, Quarterly Journal Mathematics, 48(1917), 78-92. [78] , ,k , , 28(2010), No. 1, 11-13. [79] , Gauss , , 40(2008), No. 4, ^êÆ ; 4àÜ o
B 'u g\{Ö¼êÏf¼êþŠúª X{êƆA ‘è Ü=4 g~{ÖêÏf¼êþŠìCúª °HŒÆÆg,‰Æ‡ ÿt“ ˜‡¹ ¼ê§¢ê) x²ŒÆÆ 15-18. [80] <a, ‹U, ˜‡¹ Gauss ¼ê§9Ù¢ê), p“n‰Ær, 30(2008), No. 2, 9-11. [81] Êw, o°9, 'u n ?›¥êiƒÚ¼êþŠOŽ, ÜŒÆÆ (g,‰Æ ‡), 32(2002), No. 4, 361-366. [82] , 'u n ?›¥êiÚ¼êngþŠ, —„pÄ:‰ÆÆ, 14(2001), No. 4, 286-288. [83] Êw, o°9, 'u n ?›9Ùk'OŽ¼ê, X{êƆA^êÆ, 18(2002), No. 3, 13-15. [84] Êw, 1¯, n ?›¥š"êi겐ڼêþŠ, S„“‰ŒÆÆ (g,‰ ÆÇ©‡), 39(2010), No. 3, 226-228. [85] «É, «J§êØ, ®: ‰Æч, 1999. 138 New Progress on Smarandache Problems Research Guo Xiaoyan Department of Mathematics, Northwest University, Xi’an, Shaanxi, 710127, P. R. China Yuan Xia Department of Mathematics, Northwest University, Xi’an, Shaanxi, 710127, P. R. China High American Press 2010