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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

Table of n, a(n) for n=1..7.

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 November 20 10:09 EST 2019. Contains 329334 sequences. (Running on oeis4.)