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

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

Author

Jonathan Sondow, Oct 16 2011

Keywords

• nonn

Comments

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.

Examples

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

Links

• 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.

Crossrefs

Cf. A197633, A197635, A197636.

Programs

Mathematica

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 *)

Formula

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.