A128666 Least generalized Wilson prime p such that p^2 divides (n-1)!(p-n)! - (-1)^n; or 0 if no such prime exists.

5, 2, 7, 10429, 5, 11, 17

Offset: 1

Author

Alexander Adamchuk, Mar 25 2007

Keywords

• hard
• more
• nonn

Comments

Conjecture: a(n)>0 for all n.

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 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.

Generalized Wilson primes of order 2 are listed in A079853. Generalized Wilson primes of order 17 are listed in A152413.

a(9)-a(11) = {541,11,17}.

a(13) = 13.

a(15)-a(21) = {349, 31, 61, 13151527, 71, 59, 217369}.

a(24) = 47.

a(26)-a(28) = {97579, 53, 347}.

a(30)-a(37) = {137, 20981, 71, 823, 149, 71, 4902101, 71}.

a(39)-a(45) = {491, 59, 977, 1192679, 47, 3307, 61}.

a(47) = 14197.

a(49) = 149.

a(51) = 3712567.

a(53)-a(65) = {71, 2887, 137, 35677, 467, 443, 636533, 17257, 2887, 80779, 173, 237487, 1013}.

a(67)-a(76) = {523, 373, 2341, 359, 409, 14273449, 5651, 7993, 28411, 419}.

a(78) = 227.

a(80)-a(81) = {33619,173}.

a(83) = 137.

a(85)-a(86) = {983, 6601909}.

a(88) = 859.

a(90) = 2267.

a(92)-a(94) = {1489,173,6970961}.

a(97) = 453161

a(100) = 4201.

For n<100, a(n) > 1.4*10^7 is currently not known for n in { 8, 12, 14, 22, 23, 25, 29, 31, 38, 46, 48, 50, 52, 66, 77, 79, 82, 84, 87, 89, 91, 95, 96, 98, 99 }.

Links

• Eric Weisstein's World of Mathematics, Wilson Prime
• Wikipedia, Wilson prime

Crossrefs

Cf. A007540, A007619, A079853, A124405.

Formula

If it exists, a(n) >= n. a(n) = n for n in {2, 5, 13, 563, ...} = the union of prime 2 and Wilson primes A007540.

Extensions

Edited and updated by Alexander Adamchuk, Nov 06 2010

Edited and a(18), a(21), a(26), a(36), a(42), a(51), a(59), a(62), a(64), a(72), a(86), a(94), a(97) added by Max Alekseyev, Jan 29 2012

Edited by M. F. Hasler, Dec 31 2015