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 A128666 Least generalized Wilson prime p such that p^2 divides (n-1)!(p-n)! - (-1)^n; or 0 if no such prime exists. 3
 5, 2, 7, 10429, 5, 11, 17 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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 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. CROSSREFS Cf. A007540, A007619, A079853, A124405. Sequence in context: A241388 A305574 A248259 * A013674 A155975 A152956 Adjacent sequences:  A128663 A128664 A128665 * A128667 A128668 A128669 KEYWORD hard,more,nonn AUTHOR Alexander Adamchuk, Mar 25 2007 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 STATUS approved

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Last modified September 28 17:43 EDT 2020. Contains 337393 sequences. (Running on oeis4.)