In number theory, a k-hyperperfect number is a natural number n for which the equality holds, where σ(n) is the divisor function (i.e., the sum of all positive divisors of n). A hyperperfect number is a k-hyperperfect number for some integer k. Hyperperfect numbers generalize perfect numbers, which are 1-hyperperfect.[1]
The first few numbers in the sequence of k-hyperperfect numbers are 6, 21, 28, 301, 325, 496, 697, ... (sequence A034897 in the OEIS), with the corresponding values of k being 1, 2, 1, 6, 3, 1, 12, ... (sequence A034898 in the OEIS). The first few k-hyperperfect numbers that are not perfect are 21, 301, 325, 697, 1333, ... (sequence A007592 in the OEIS).
List of hyperperfect numbers
editThe following table lists the first few k-hyperperfect numbers for some values of k, together with the sequence number in the On-Line Encyclopedia of Integer Sequences (OEIS) of the sequence of k-hyperperfect numbers:
k | k-hyperperfect numbers | OEIS |
---|---|---|
1 | 6, 28, 496, 8128, 33550336, ... | OEIS: A000396 |
2 | 21, 2133, 19521, 176661, 129127041, ... | OEIS: A007593 |
3 | 325, ... | |
4 | 1950625, 1220640625, ... | |
6 | 301, 16513, 60110701, 1977225901, ... | OEIS: A028499 |
10 | 159841, ... | |
11 | 10693, ... | |
12 | 697, 2041, 1570153, 62722153, 10604156641, 13544168521, ... | OEIS: A028500 |
18 | 1333, 1909, 2469601, 893748277, ... | OEIS: A028501 |
19 | 51301, ... | |
30 | 3901, 28600321, ... | |
31 | 214273, ... | |
35 | 306181, ... | |
40 | 115788961, ... | |
48 | 26977, 9560844577, ... | |
59 | 1433701, ... | |
60 | 24601, ... | |
66 | 296341, ... | |
75 | 2924101, ... | |
78 | 486877, ... | |
91 | 5199013, ... | |
100 | 10509080401, ... | |
108 | 275833, ... | |
126 | 12161963773, ... | |
132 | 96361, 130153, 495529, ... | |
136 | 156276648817, ... | |
138 | 46727970517, 51886178401, ... | |
140 | 1118457481, ... | |
168 | 250321, ... | |
174 | 7744461466717, ... | |
180 | 12211188308281, ... | |
190 | 1167773821, ... | |
192 | 163201, 137008036993, ... | |
198 | 1564317613, ... | |
206 | 626946794653, 54114833564509, ... | |
222 | 348231627849277, ... | |
228 | 391854937, 102744892633, 3710434289467, ... | |
252 | 389593, 1218260233, ... | |
276 | 72315968283289, ... | |
282 | 8898807853477, ... | |
296 | 444574821937, ... | |
342 | 542413, 26199602893, ... | |
348 | 66239465233897, ... | |
350 | 140460782701, ... | |
360 | 23911458481, ... | |
366 | 808861, ... | |
372 | 2469439417, ... | |
396 | 8432772615433, ... | |
402 | 8942902453, 813535908179653, ... | |
408 | 1238906223697, ... | |
414 | 8062678298557, ... | |
430 | 124528653669661, ... | |
438 | 6287557453, ... | |
480 | 1324790832961, ... | |
522 | 723378252872773, 106049331638192773, ... | |
546 | 211125067071829, ... | |
570 | 1345711391461, 5810517340434661, ... | |
660 | 13786783637881, ... | |
672 | 142718568339485377, ... | |
684 | 154643791177, ... | |
774 | 8695993590900027, ... | |
810 | 5646270598021, ... | |
814 | 31571188513, ... | |
816 | 31571188513, ... | |
820 | 1119337766869561, ... | |
968 | 52335185632753, ... | |
972 | 289085338292617, ... | |
978 | 60246544949557, ... | |
1050 | 64169172901, ... | |
1410 | 80293806421, ... | |
2772 | 95295817, 124035913, ... | OEIS: A028502 |
3918 | 61442077, 217033693, 12059549149, 60174845917, ... | |
9222 | 404458477, 3426618541, 8983131757, 13027827181, ... | |
9828 | 432373033, 2797540201, 3777981481, 13197765673, ... | |
14280 | 848374801, 2324355601, 4390957201, 16498569361, ... | |
23730 | 2288948341, 3102982261, 6861054901, 30897836341, ... | |
31752 | 4660241041, 7220722321, 12994506001, 52929885457, 60771359377, ... | OEIS: A034916 |
55848 | 15166641361, 44783952721, 67623550801, ... | |
67782 | 18407557741, 18444431149, 34939858669, ... | |
92568 | 50611924273, 64781493169, 84213367729, ... | |
100932 | 50969246953, 53192980777, 82145123113, ... |
It can be shown that if k > 1 is an odd integer and and are prime numbers, then is k-hyperperfect; Judson S. McCranie has conjectured in 2000 that all k-hyperperfect numbers for odd k > 1 are of this form, but the hypothesis has not been proven so far. Furthermore, it can be proven that if p ≠ q are odd primes and k is an integer such that then pq is k-hyperperfect.
It is also possible to show that if k > 0 and is prime, then for all i > 1 such that is prime, is k-hyperperfect. The following table lists known values of k and corresponding values of i for which n is k-hyperperfect:
k | Values of i | OEIS |
---|---|---|
16 | 11, 21, 127, 149, 469, ... | OEIS: A034922 |
22 | 17, 61, 445, ... | |
28 | 33, 89, 101, ... | |
36 | 67, 95, 341, ... | |
42 | 4, 6, 42, 64, 65, ... | OEIS: A034923 |
46 | 5, 11, 13, 53, 115, ... | OEIS: A034924 |
52 | 21, 173, ... | |
58 | 11, 117, ... | |
72 | 21, 49, ... | |
88 | 9, 41, 51, 109, 483, ... | OEIS: A034925 |
96 | 6, 11, 34, ... | |
100 | 3, 7, 9, 19, 29, 99, 145, ... | OEIS: A034926 |
Hyperdeficiency
editThe newly introduced mathematical concept of hyperdeficiency is related to the hyperperfect numbers.
Definition (Minoli 2010): For any integer n and for integer k > 0, define the k-hyperdeficiency (or simply the hyperdeficiency) for the number n as
A number n is said to be k-hyperdeficient if
Note that for k = 1 one gets which is the standard traditional definition of deficiency.
Lemma: A number n is k-hyperperfect (including k = 1) if and only if the k-hyperdeficiency of n,
Lemma: A number n is k-hyperperfect (including k = 1) if and only if for some k, for at least one j > 0.
References
edit- ^ Weisstein, Eric W. "Hyperperfect Number". mathworld.wolfram.com. Retrieved 2020-08-10.
- Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. p. 114. ISBN 1-4020-4215-9. Zbl 1151.11300.
Further reading
editArticles
edit- Minoli, Daniel; Bear, Robert (Fall 1975), "Hyperperfect numbers", Pi Mu Epsilon Journal, 6 (3): 153–157.
- Minoli, Daniel (Dec 1978), "Sufficient forms for generalized perfect numbers", Annales de la Faculté des Sciences UNAZA, 4 (2): 277–302.
- Minoli, Daniel (Feb 1981), "Structural issues for hyperperfect numbers", Fibonacci Quarterly, 19 (1): 6–14, doi:10.1080/00150517.1981.12430116.
- Minoli, Daniel (April 1980), "Issues in non-linear hyperperfect numbers", Mathematics of Computation, 34 (150): 639–645, doi:10.2307/2006107, JSTOR 2006107.
- Minoli, Daniel (October 1980), "New results for hyperperfect numbers", Abstracts of the American Mathematical Society, 1 (6): 561.
- Minoli, Daniel; Nakamine, W. (1980). "Mersenne numbers rooted on 3 for number theoretic transforms". ICASSP '80. IEEE International Conference on Acoustics, Speech, and Signal Processing. Vol. 5. pp. 243–247. doi:10.1109/ICASSP.1980.1170906..
- McCranie, Judson S. (2000), "A study of hyperperfect numbers", Journal of Integer Sequences, 3: 13, Bibcode:2000JIntS...3...13M, archived from the original on 2004-04-05.
- te Riele, Herman J.J. (1981), "Hyperperfect numbers with three different prime factors", Math. Comp., 36 (153): 297–298, doi:10.1090/s0025-5718-1981-0595066-9, MR 0595066, Zbl 0452.10005.
- te Riele, Herman J.J. (1984), "Rules for constructing hyperperfect numbers", Fibonacci Q., 22: 50–60, doi:10.1080/00150517.1984.12429920, Zbl 0531.10005.
Books
edit- Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p. 114-134)