A152413 Generalized Wilson primes of order 17; or primes p such that p^2 divides 16!(p-17)! + 1.

61, 251, 479

Offset

1,1

Comments

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.

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).

Links

Table of n, a(n) for n=1..3.

Eric Weisstein's World of Mathematics, Wilson Prime

Crossrefs

Cf. A007540, A007619, A079853, A124405, A128666.

Sequence in context: A158673 A174333 A158680 * A029815 A142424 A252803

Adjacent sequences: A152410 A152411 A152412 * A152414 A152415 A152416

Keyword

bref,hard,more,nonn

Author

Alexander Adamchuk, Dec 03 2008

Extensions

Edited by Max Alekseyev, Jan 28 2012

Status

approved