%I #23 Jul 03 2024 12:38:16

%S 1,2,4,5,8,10

%N Numbers k such that (prime(k)-1)! + prime(k)^8 is prime.

%C k = {1, 2, 4, 5, 8, 10} yields primes p(k) = {2, 3, 7, 11, 19, 29}. There are no more such k up to k=100. Computed in collaboration with _Ray Chandler_.

%C a(7) > 600. - _Jinyuan Wang_, Apr 10 2020

%C a(7) > 900. - _Tyler NeSmith_, May 05 2021

%C a(7) > 2500. - _Michael S. Branicky_, Jul 03 2024

%e a(1) = 1 because (prime(1)-1)! + prime(1)^8 = (2-1)! + 2^8 = 257 is the smallest prime of that form.

%e a(5) = 8 because (prime(8)-1)! + prime(8)^8 = (19-1)! + 19^8 = 304888344611713861001750412961 is the 5th smallest prime of that form.

%t lst={};Do[p=Prime[n];If[PrimeQ[(p-1)!+p^8], AppendTo[lst, n]], {n, 10^2}];lst (* _Vladimir Joseph Stephan Orlovsky_, Sep 08 2008 *)

%o (PARI) is(k) = ispseudoprime((prime(k)-1)! + prime(k)^8); \\ _Jinyuan Wang_, Apr 10 2020

%Y Cf. A100595, A100599, A100858.

%K nonn,hard,more,less

%O 1,2

%A _Jonathan Vos Post_, Nov 30 2004

%E Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jun 08 2007