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%I #29 Feb 11 2019 16:53:43
%S 5,2,7,10429,5,11,17
%N Least generalized Wilson prime p such that p^2 divides (n-1)!(p-n)! - (-1)^n; or 0 if no such prime exists.
%C Conjecture: a(n)>0 for all n.
%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 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.
%C Generalized Wilson primes of order 2 are listed in A079853. Generalized Wilson primes of order 17 are listed in A152413.
%C a(9)-a(11) = {541,11,17}.
%C a(13) = 13.
%C a(15)-a(21) = {349, 31, 61, 13151527, 71, 59, 217369}.
%C a(24) = 47.
%C a(26)-a(28) = {97579, 53, 347}.
%C a(30)-a(37) = {137, 20981, 71, 823, 149, 71, 4902101, 71}.
%C a(39)-a(45) = {491, 59, 977, 1192679, 47, 3307, 61}.
%C a(47) = 14197.
%C a(49) = 149.
%C a(51) = 3712567.
%C a(53)-a(65) = {71, 2887, 137, 35677, 467, 443, 636533, 17257, 2887, 80779, 173, 237487, 1013}.
%C a(67)-a(76) = {523, 373, 2341, 359, 409, 14273449, 5651, 7993, 28411, 419}.
%C a(78) = 227.
%C a(80)-a(81) = {33619,173}.
%C a(83) = 137.
%C a(85)-a(86) = {983, 6601909}.
%C a(88) = 859.
%C a(90) = 2267.
%C a(92)-a(94) = {1489,173,6970961}.
%C a(97) = 453161
%C a(100) = 4201.
%C 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 }.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WilsonPrime.html">Wilson Prime</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Wilson_prime">Wilson prime</a>
%F 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.
%Y Cf. A007540, A007619, A079853, A124405.
%K hard,more,nonn
%O 1,1
%A _Alexander Adamchuk_, Mar 25 2007
%E Edited and updated by _Alexander Adamchuk_, Nov 06 2010
%E 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
%E Edited by _M. F. Hasler_, Dec 31 2015
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