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

1,3

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.

Links

Table of n, a(n) for n=1..63.

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.

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.

Example

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

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 *)

Crossrefs

Cf. A197633, A197635, A197636.

Sequence in context: A011346 A155525 A019741 * A011482 A259929 A053007

Adjacent sequences: A197631 A197632 A197633 * A197635 A197636 A197637

Keyword

nonn

Author

Jonathan Sondow, Oct 16 2011

Status

approved