%I #21 Sep 13 2022 08:33:44
%S 4,6,10,28,772,1320,2620
%N Numbers k such that (k!+1)/(k+1) is prime.
%C Subsequence of A006093.
%C It is unknown whether this sequence is infinite. If S = (n!+k)/(n+k), when k > 1, there are finitely many nvalues that make S an integer. When k = 1, there are infinitely many nvalues that make S an integer (A006093).
%C a(8) > 4725.
%C If it exists, a(8) > 6778.  _J.W.L. (Jan) Eerland_, Aug 15 2022
%e (4!+1)/(4+1) = 5 is prime. Thus, 4 is a member of this sequence.
%t ParallelTable[If[PrimeQ[(Factorial[n]+1)/(n+1)],n,Nothing],{n,1,5*10^3}]//.{}>Nothing (* _J.W.L. (Jan) Eerland_, Aug 15 2022 *)
%o (PARI) for(n=1,5000,s=((prime(n)1)!+1)/(prime(n)); if(ispseudoprime(s), print1(prime(n)1)))
%Y Cf. A006093.
%K nonn,more,hard
%O 1,1
%A _Derek Orr_, May 17 2014
