A197635 -id:A197635 - cp's OEIS frontend

A197636 Non-Wilson primes: primes p such that (p-1)! =/= -1 mod p^2.

2, 3, 7, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 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

Offset: 1

Jonathan Sondow, Oct 19 2011

nonn

All primes except 5, 13, 563, and any other Wilson prime A007540 that may exist.
Same as primes p that do not divide their Wilson quotient ((p-1)!+1)/p.
Wilson's theorem says that (p-1)! == -1 (mod p) if and only if p is prime.
p = prime(i) is a term iff A250406(i) != 0. - Felix Fröhlich, Jan 24 2016
Complement of A007540 in A000040. - Felix Fröhlich, Jan 24 2016

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

E. Costa, R. Gerbicz and D. Harvey, A search for Wilson primes, Mathematics of Computation, 83 (2014), 3071-3091 (arXiv:1209.3436 [math.NT], 2012).
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, in Proceedings of CANT 2011, arXiv:1110.3113 [math.NT], 2011-2012.
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.

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

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

forprime(p=1, 500, if(Mod((p-1)!, p^2)!=-1, print1(p, ", "))) \\ Felix Fröhlich, Jan 24 2016

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

A197633 Fermat-Wilson quotients of non-Wilson primes: q_p(w_p), where q_p(k) = (k^(p-1)-1)/p is a Fermat quotient, w_p = ((p-1)!+1)/p is a Wilson quotient, and p is a non-Wilson prime.

Original entry on oeis.org

0, 0, 170578899504, 1387752405580695978098914368989316131852701063520729400

Offset: 1

Jonathan Sondow, Oct 16 2011

nonn

A Wilson prime is a prime p that divides its Wilson quotient w_p (see A007619). The known Wilson primes are 5, 13, 563 (see A007540).
If p is a non-Wilson prime (see A197636), then p does not divide w_p, and so by Fermat's little theorem the Fermat quotient q_p(w_p) is an integer.
The next term is the Fermat-Wilson quotient of 17, which has 193 digits.
The Fermat-Wilson quotient of 14771 (see A197635) has over 800 million digits.
The GCD of all Fermat-Wilson quotients is 24. In particular, q_p(w_p) is never prime.

			The 3rd non-Wilson prime is 7, so a(3) = (((6!+1)/7)^6-1)/7 = 170578899504.

Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, in Proceedings of CANT 2011, arXiv:1110.3113
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.

Cf. A007540, A007619, A197634, A197635, A197636, A216867.

nmax=4; nonWilsonQ[p_] := Mod[((p-1)!+1)/p ,p] != 0; A197636 = Select[ Prime[ Range[nmax+2]], nonWilsonQ]; a[n_] := With[{p=A197636[[n]]}, ((((p-1)!+1)/p)^(p-1)-1)/p]; Table[ a[n], {n, 1, nmax}] (* Jean-François Alcover, Dec 14 2011 *)

a(n) = ((((p-1)!+1)/p)^(p-1)-1)/p, where p = A197636(n).

A197634 Fermat-Wilson remainders: Fermat-Wilson quotients A197633 of non-Wilson primes p modulo p.

Original entry on oeis.org

0, 0, 6, 7, 9, 7, 1, 6, 18, 17, 30, 11, 25, 30, 24, 46, 64, 16, 18, 4, 29, 66, 95, 11, 10, 9, 8, 64, 118, 77, 136, 15, 139, 64, 105, 9, 153, 167, 93, 4, 144, 180, 67, 179, 133, 51, 145, 130, 168, 41, 25, 163, 51, 42, 43, 100, 162, 212, 235, 2, 98, 232, 22

Offset: 1

Jonathan Sondow, Oct 16 2011

nonn

a(n) = 0 iff the n-th non-Wilson prime is a Wieferich-non-Wilson prime A197635. For example a(1) = a(2) = 0, so the 1st and 2nd non-Wilson primes 2 and 3 are Wieferich-non-Wilson primes. The next one is 14771, which is the 1728th non-Wilson prime, so the next zero in the sequence occurs at a(1728) = 0.

			The 3rd non-Wilson prime is 7, and A197633(3) = 170578899504 == 6 (mod 7), so a(3) = 6.

Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, in Proceedings of CANT 2011, arXiv:1110.3113
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.

Cf. A197633, A197635, A197636.

nmax = 63; nonWilsonQ[p_] := Mod[((p-1)! + 1)/p, p] != 0; nwp = Select[ Prime[ Range[nmax + 2]], nonWilsonQ]; A197633[n_] := With[{p = nwp[[n]]}, ((((p-1)! + 1)/p)^(p-1) - 1)/p]; a[n_] := Mod[A197633[n], nwp[[n]]]; Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Oct 10 2012 *)

a(n) = A197633(n) mod A197636(n).

a(n) = ((((p-1)!+1)/p)^(p-1)-1)/p mod p, where p is the n-th non-Wilson prime.

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A197636 Non-Wilson primes: primes p such that (p-1)! =/= -1 mod p^2.

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A197633 Fermat-Wilson quotients of non-Wilson primes: q_p(w_p), where q_p(k) = (k^(p-1)-1)/p is a Fermat quotient, w_p = ((p-1)!+1)/p is a Wilson quotient, and p is a non-Wilson prime.

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A197634 Fermat-Wilson remainders: Fermat-Wilson quotients A197633 of non-Wilson primes p modulo p.

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