A018239 Primorial primes: primes of the form 1 + product of first k primes, for some k.

2, 3, 7, 31, 211, 2311, 200560490131

Offset: 1

Author

Murray R. Bremner

Keywords

• nonn
• nice

Comments

Prime numbers that are the sum of two primorial numbers. - Juri-Stepan Gerasimov, Nov 08 2010

Examples

			From _M. F. Hasler_, Jun 23 2019: (Start)
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)
		

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

Crossrefs

Primes in A006862 (primorials plus 1).

A005234 and A014545 (which are the main entries for this sequence) give more terms.

Cf. A002110.

Programs

Mathematica

Select[FoldList[Times, 1, Prime[Range[200]]] + 1, PrimeQ] (* Loreno Heer (helohe(AT)bluewin.ch), Jun 29 2004 *)

PARI

P=1;print1(2); forprime(p=2,1e6, if(isprime(1+P*=p), print1(", "P+1))) \\ Charles R Greathouse IV, Apr 28 2015

Formula

a(n) = 1 + A002110(A014545(n)), where A002110(k) is the product of the first k primes. - M. F. Hasler, Jun 23 2019

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