1,1

By Wilson's Theorem, for prime p the Wilson quotient ((p-1)!+1)/p is an integer A007619. By Gauss's extension (see Dickson p. 65), the generalized Wilson quotient (P(n)+e(n))/n is an integer, where P(n) = n-phi-torial A001783 and e(n) = +1 or -1 according as n does or does not have a primitive root (see A033948).

For additional references and links, see A007540.

T. Agoh, K. Dilcher, and L. Skula, Wilson quotients for composite moduli, Math. Comp. 67 (1998), 843-861.

K. E. Kloss, Some number theoretic calculations, J. Res. Nat. Bureau of Stand., B, 69 (1965), 335-339.

L. E. Dickson, History of the Theory of Numbers, vol. 1, Divisibility and Primality, Chelsea, New York, 1966.

T. Agoh, K. Dilcher, and L. Skula, Wilson quotients for composite moduli.

a(n) = (P(n)+e(n))/n, with P(n) = n-phi-torial = A001783(n) and e(n) = +1 if n = 1, 2, 4, p^k or 2p^k, where p is an odd prime and k > 0, and e(n) = -1 otherwise.

P(8) = 3*5*7 = 105 and e(8) = -1, so a(8) = (105-1)/8 = 13.

Contribution from Peter Luschny (peter(AT)luschny.de), Jul 19 2009: (Start)

a := proc(n) local A001783, e, i;

A001783 := proc(n) local i; mul(i, i=select(k->igcd(k, n)=1, [$1..n]))end;

e := proc(n) local p, r, P; if n=1 or n=2 or n=4 then RETURN(1) fi;

P := select(isprime, [$3..n]); for p in P do r := p;

while r <= n do if n = r or n = 2*r then RETURN(1) fi;

r := r*p; od od; -1 end; (A001783(n)+e(n))/n end: (End)

Cf. Wilson quotient A007619, Wilson prime A007540, Wilson number A157250, n-phi-torial A001783, numbers having a primitive root A033948.

Sequence in context: A014651 A082063 A099940 this_sequence A155586 A069739 A066060

Adjacent sequences: A157246 A157247 A157248 this_sequence A157250 A157251 A157252

nonn

Jonathan Sondow (jsondow(AT)alumni.princeton.edu) and Wadim Zudilin (wzudilin(AT)mpim-bonn.mpg.de), Feb 27 2009