A100603 Numbers k such that (prime(k)-1)! + prime(k)^4 is prime.

1, 2, 4, 67, 212, 1615, 2570

Offset: 1

Author

Jonathan Vos Post, Nov 30 2004

Keywords

• nonn
• more

Comments

k = {1, 2, 4, 67} yields primes p(k) = {2, 3, 7, 331}. There are no more such k up to k=100. Computed in collaboration with Ray Chandler.

Terms a(5) and greater are only probable primes. - Iain Fox, Mar 05 2018

Examples

			a(3) = 4 because (prime(4)-1)! + prime(4)^4 = (7-1)! + 7^4 = 720 + 2401 = 3121 is the 3rd prime of this form.
		

Crossrefs

Cf. A100858.

Programs

Mathematica

lst={};Do[p=Prime[n];If[PrimeQ[(p-1)!+p^4], AppendTo[lst, n]], {n, 10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 08 2008 *)

PARI

is(k) = my(p=prime(k)); ispseudoprime((p-1)! + p^4) \\ Iain Fox, Mar 05 2018

Formula

Numbers k such that (prime(k)-1)! + prime(k)^4 is prime, where prime(k) is the k-th prime.

Extensions

a(5) from Iain Fox, Mar 05 2018

a(6) from Iain Fox, Mar 11 2018

a(7) from Michael S. Branicky, May 29 2025