%I #18 Feb 16 2025 08:33:09
%S 61,251,479
%N Generalized Wilson primes of order 17; or primes p such that p^2 divides 16!(p-17)! + 1.
%C Wilson's theorem states that (p-1)! == -1 (mod p) for every prime p. Wilson primes are the primes p such that p^2 divides (p-1)! + 1. They are listed in A007540. Wilson's theorem can be expressed in general as (n-1)!(p-n)! == (-1)^n (mod p) for every prime p >= n. Generalized Wilson primes order n are the primes p such that p^2 divides (n-1)!(p-n)! - (-1)^n.
%C Alternatively, prime p=prime(k) is a generalized Wilson prime order n iff A002068(k) == A007619(k) == H(n-1) (mod p), where H(n-1) = A001008(n-1)/A002805(n-1) is (n-1)-st harmonic number. For this sequence (n=17), it reduces to A002068(k) == A007619(k) == 2436559/720720 (mod p).
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WilsonPrime.html">Wilson Prime</a>
%Y Cf. A007540, A007619, A079853, A124405, A128666.
%K bref,hard,more,nonn,changed
%O 1,1
%A _Alexander Adamchuk_, Dec 03 2008
%E Edited by _Max Alekseyev_, Jan 28 2012