A051888 a(n) is the smallest prime p such that p*n! + 1 is prime.

2, 2, 2, 2, 3, 2, 3, 3, 7, 3, 3, 5, 2, 3, 13, 7, 31, 5, 2, 7, 17, 67, 41, 3, 13, 3, 43, 17, 97, 7, 29, 109, 3, 71, 5, 2, 7, 41, 3, 59, 3, 11, 29, 7, 107, 67, 79, 3, 743, 149, 163, 2, 211, 2, 19, 71, 73, 23, 37, 113, 149, 67, 41, 617, 107, 37, 107, 283, 113, 19, 239, 107, 73, 97, 5

Offset: 0

Author

Labos Elemer, Dec 15 1999

Keywords

• nonn

Comments

Analogous to or subset of A051686; generalization of A005384.

The PFGW program has been used to certify all the primes corresponding to the terms up to a(1000), using a deterministic test which exploits the factorization of a(n) - 1. - Giovanni Resta, May 30 2018

Links

• Giovanni Resta, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A005384, A051686, A051886, A051887, A051901, A064983.

Programs

Mathematica

Do[k = 1; While[ !PrimeQ[ Prime[k]*n! + 1], k++ ]; Print[ Prime[k]], {n, 1, 75} ]
spp[n_]:=Module[{p=2,nf=n!},While[!PrimeQ[p*nf+1],p=NextPrime[p]];p]; Array[ spp,80,0] (* Harvey P. Dale, May 17 2019 *)

PARI

a(n) = {my(p=2); while (!isprime(p*n! + 1), p = nextprime(p+1)); p;} \\ Michel Marcus, May 28 2018

Formula

a(n) = (A051901(n)-1)/n!. - Amiram Eldar, Feb 25 2025

Extensions

More terms from James Sellers, Dec 16 1999