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COMMENTS
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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 }.
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