OFFSET
1,1
COMMENTS
Prime numbers that are the sum of two primorial numbers. - Juri-Stepan Gerasimov, Nov 08 2010
REFERENCES
F. Iacobescu, Smarandache Partition Type and Other Sequences, Bull. Pure Appl. Sciences, Vol. 16E, No. 2 (1997), pp. 237-240.
LINKS
Michel Marcus, Table of n, a(n) for n = 1..10
M. Fleuren, Factors and primes of Smarandache sequences.
M. Fleuren, Smarandache Prime Product Sequence.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2023. - From N. J. A. Sloane, Jun 13 2012
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.
R. Ondrejka, Primorial-Plus-One Primes
FORMULA
a(n) = 1 + A002110(A014545(n)), where A002110(k) is the product of the first k primes. - M. F. Hasler, Jun 23 2019
EXAMPLE
From M. F. Hasler, Jun 23 2019:
a(1) = 2 = 1 + product of the first 0 primes (i.e., the empty product = 1).
a(2) = 3 = 1 + 2 = 1 + product of the first prime (= 2).
a(3) = 7 = 1 + 2*3 = 1 + product of the first two primes.
a(4) = 31 = 1 + 2*3*5 = 1 + product of the first three primes.
a(5) = 211 = 1 + 2*3*5*7 = 1 + product of the first four primes.
a(6) = 2311 = 1 + 2*3*5*7*11 = 1 + product of the first five primes.
Then the product of the first 6, 7, ..., 9 or 10 primes does not yield a primorial prime, the next one is:
a(7) = 200560490131 = 1 + 2*3*5*7*11*13*17*19*23*29*31 = 1 + product of the first eleven primes,
and so on. See A014545 = (0, 1, 2, 3, 4, 5, 11, 75, 171, 172, ...) for the k's that yield a term. (End)
MATHEMATICA
Select[FoldList[Times, 1, Prime[Range[200]]] + 1, PrimeQ] (* Loreno Heer (helohe(AT)bluewin.ch), Jun 29 2004 *)
PROG
(PARI) P=1; print1(2); forprime(p=2, 1e6, if(isprime(1+P*=p), print1(", "P+1))) \\ Charles R Greathouse IV, Apr 28 2015
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane at the suggestion of Vladeta Jovovic, Jun 18 2007
Name edited by M. F. Hasler, Jun 23 2019
STATUS
approved