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.

0, 0, 170578899504, 1387752405580695978098914368989316131852701063520729400

Offset

1,3

Comments

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.

Links

Table of n, a(n) for n=1..4.

J. 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

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

Example

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

Mathematica

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 *)

Crossrefs

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

Sequence in context: A304235 A233503 A241501 * A339122 A105295 A288262

Adjacent sequences: A197630 A197631 A197632 * A197634 A197635 A197636

Keyword

nonn

Author

Jonathan Sondow, Oct 16 2011

Status

approved