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COMMENT
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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 = {5, 13, 563}. 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. For n = 2 generalized Wilson primes are listed in A079853 = {2, 3, 11, 107, 4931} Primes p such that (p-2)! == 1 mod p^2. a(n) >= n. a(n) = n for n = {2, 5, 13, 563, ...} = a union of prime 2 and Wilson primes {5, 13, 563, ...}. a(9)-a(11) = {541,11,17}. a(13) = 13. a(15)-a(17) = {349,31,61}. a(19)-a(20) = {71,59}. a(24) = 47. a(27)-a(28) = {53,347}. a(30) = 137. a(32)-a(35) = {71,823,149,71}. a(37) = 71. a(39)-a(41) = {491,59,977}. a(43)-a(45) = {47,3307,61}. a(47) = 14197. a(49) = 149. a(53)-a(54) = {71,2887}. a(n)>Prime(2000) is currently not known for n = {8,12,14,18,21,22,23,25,26,29,31,36,38,42,46,48,50,51,52,...}.
Comments from Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 16 2008 (Start): a(9)-a(11) = {541,11,17}.
a(13) = 13.
a(15)-a(17) = {349,31,61}.
a(19)-a(20) = {71,59}.
a(24) = 47.
a(27)-a(28) = {53,347}.
a(30)-a(35) = {137,20981,71,823,149,71}.
a(37) = 71.
a(39)-a(41) = {491,59,977}.
a(43)-a(45) = {47,3307,61}.
a(47) = 14197.
a(49) = 149.
a(53)-a(58) = {71,2887,137,35677,467,443}.
a(60)-a(61) = {17257,2887}.
a(63)= 173.
a(65) = 1013.
a(67)-a(71) = {523,373,2341,359,409}.
a(73)-a(76) = {5651,7993,28411, 419}.
a(78) = 227.
a(80)-a(81) = {33619,173}.
a(83) = 137.
a(85) = 983.
a(88) = 859.
a(90) = 2267.
a(92)-a(93) = {1489,173}.
a(100) = 4201.
a(n)>prime(6000) is currently not known for n = {8,12,14,18,21-23,25,26,29,31,36,38,42,46,48,50-52,59,62,64,66,72,77,79,82,84,86,87,89,91,94-99,...}. (End)
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