OFFSET
1,1
COMMENTS
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.
REFERENCES
L. E. Dickson, History of the Theory of Numbers, vol. 1, Divisibility and Primality, Chelsea, New York, 1966.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..200
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. Research of the National Bureau of Standards-B. Mathematics and Mathematical Physics, Vol. 69B, No. 4 (1965), 335-336.
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 [math.NT], 2011-2012.
FORMULA
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.
EXAMPLE
P(8) = 3*5*7 = 105 and e(8) = -1, so a(8) = (105-1)/8 = 13.
MAPLE
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:
# Peter Luschny, Jul 19 2009
MATHEMATICA
p[n_] := Times @@ Select[ Range[n], CoprimeQ[n, #] & ]; e[1 | 2 | 4] = 1; e[n_] := (fi = FactorInteger[n]; If[MatchQ[fi, {{(p_)?OddQ, _}} | {{2, 1}, {_, _}}], 1, -1]); a[n_] := (p[n] + e[n])/n; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Sep 28 2011 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Jonathan Sondow and Wadim Zudilin, Feb 27 2009
STATUS
approved