A197636 - OEIS

%I #29 May 21 2016 22:49:32

%S 2,3,7,11,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,

%T 101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,

%U 191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,523,541,547,557,569

%N Non-Wilson primes: primes p such that (p-1)! =/= -1 mod p^2.

%C All primes except 5, 13, 563, and any other Wilson prime A007540 that may exist.

%C Same as primes p that do not divide their Wilson quotient ((p-1)!+1)/p.

%C Wilson's theorem says that (p-1)! == -1 (mod p) if and only if p is prime.

%C p = prime(i) is a term iff A250406(i) != 0. - _Felix Fröhlich_, Jan 24 2016

%C Complement of A007540 in A000040. - _Felix Fröhlich_, Jan 24 2016

%H E. Costa, R. Gerbicz and D. Harvey, <a href="http://dx.doi.org/10.1090/S0025-5718-2014-02800-7">A search for Wilson primes</a>, Mathematics of Computation, 83 (2014), 3071-3091 (arXiv:<a href="http://arxiv.org/abs/1209.3436">1209.3436</a> [math.NT], 2012).

%H J. Sondow, <a href="http://arxiv.org/abs/1110.3113"> Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771</a>, in Proceedings of CANT 2011, arXiv:1110.3113 [math.NT], 2011-2012.

%H J. Sondow, <a href="http://dx.doi.org/10.1007/978-1-4939-1601-6_17">Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771</a>, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.

%F ((p-1)!+1)/p =/= 0 (mod p), where p is prime.

%e 2 is a non-Wilson prime since (2-1)! = 1 ==/== -1 (mod 2^2).

%t Select[Prime@ Range@ 104, Mod[Factorial[# - 1], #^2] != #^2 - 1 &] (* _Michael De Vlieger_, Jan 24 2016 *)

%o (PARI) forprime(p=1, 500, if(Mod((p-1)!, p^2)!=-1, print1(p, ", "))) \\ _Felix Fröhlich_, Jan 24 2016

%Y Cf. A007540, A007619, A197633, A197634, A197635, A250406.

%K nonn

%O 1,1

%A _Jonathan Sondow_, Oct 19 2011