A301373 Numbers k such that (k+1)!*k/2 + 1 is prime.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 19, 24, 251, 374, 953, 1104, 1507, 3390, 4443, 5762

Offset: 1

Author

Daniel Suteu, Apr 03 2018

Keywords

• nonn
• more

Comments

The associated primes are A300559(a(n)) = A180119(a(n))+1 = A001286(a(n)+1)+1. - M. F. Hasler, Apr 10 2018

Looking for primes of the form p(n) = 1 + n! f(n) with a simple polynomial function f, it appears that the choice f(n) = n(n+1)/2 = A000217 is one of the most successful choices for getting a maximum of primes for n = 1..20. - M. F. Hasler, Apr 14 2018

The PFGW program has been used to certify all the terms up to a(23), using a deterministic test which exploits the factorization of a(n) - 1. - Giovanni Resta, Jun 24 2018

Links

• Maheswara Rao Valluri, Primes of the form p = 1 + n! Sum n, for some n ∈ N*, arXiv:1803.11461 [math.GM], 2018.

Crossrefs

Cf. A090703, A300559, A180119, A001286.

See A302859 for the actual primes.

Programs

Mathematica

Do[ If[ PrimeQ[n(n +1)!/2 +1], Print@ n], {n, 4000}] (* Robert G. Wilson v, Apr 05 2018 *)

PARI

isok(k) = ispseudoprime((k+1)! * k / 2 + 1);

Extensions

a(21) from Robert G. Wilson v, Apr 05 2018

a(22) from Vaclav Kotesovec, Apr 06 2018

a(23) from Giovanni Resta, Jun 24 2018