%N Numbers n such that (n!-k)/(n-k) is prime for some k.
%C a(14) > 3000.
%C There is only one value of k that can work, and it is n/2. Thus, all members of the sequence are even.
%C Even numbers n such that 2*(n-1)!-1 is prime.
%C (Odd members of A076133) + 1. - _Robert Israel_, Aug 11 2016
%e (4!-k)/(4-k) is prime for some k (namely, k = 2). Thus, 4 is a member of this sequence.
%p select(t -> isprime(2*(t-1)!-1), [seq(q,q=2..1000, 2)]); # _Robert Israel_, Aug 11 2016
%t Select[Range[2, 10^3, 2], PrimeQ[2 (# - 1)! - 1] &] (* _Michael De Vlieger_, Aug 11 2016 *)
%o (PARI) a(n)=for(k=1,int(n/2),s=(n!-k)/(n-k);if(floor(s)==s,if(ispseudoprime(s),return(k))))
%o n=1;while(n<1000,if(a(n),print1(n,", "));n+=1)
%Y Cf. A076133.
%A _Derek Orr_, May 26 2014
%E Edited and a(14)-a(16) added by _Robert Israel_, Aug 11 2016