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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.

%I A197633 #35 Nov 08 2024 08:01:39
%S A197633 0,0,170578899504,
%T A197633 1387752405580695978098914368989316131852701063520729400
%N 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.
%C A197633 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).
%C A197633 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.
%C A197633 The next term is the Fermat-Wilson quotient of 17, which has 193 digits.
%C A197633 The Fermat-Wilson quotient of 14771 (see A197635) has over 800 million digits.
%C A197633 The GCD of all Fermat-Wilson quotients is 24. In particular, q_p(w_p) is never prime.
%H A197633 Jonathan 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
%H A197633 Jonathan Sondow, <a href="http://link.springer.com/chapter/10.1007%2F978-1-4939-1601-6_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 A197633 a(n) = ((((p-1)!+1)/p)^(p-1)-1)/p, where p = A197636(n).
%e A197633 The 3rd non-Wilson prime is 7, so a(3) = (((6!+1)/7)^6-1)/7 = 170578899504.
%t A197633 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 *)
%Y A197633 Cf. A007540, A007619, A197634, A197635, A197636, A216867.
%K A197633 nonn
%O A197633 1,3
%A A197633 _Jonathan Sondow_, Oct 16 2011