A157249 Generalized Wilson quotients (or Wilson quotients for composite moduli).

2, 1, 1, 1, 5, 1, 103, 13, 249, 19, 329891, 32, 36846277, 1379, 59793, 126689, 1230752346353, 4727, 336967037143579, 436486, 2252263619, 56815333, 48869596859895986087, 1549256, 1654529071288638505, 23390099351, 56463097772562963, 51860555558, 10513391193507374500051862069

Offset: 1

Author

Jonathan Sondow and Wadim Zudilin, Feb 27 2009

Keywords

• nonn

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.

Examples

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

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.

Crossrefs

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

Cf. A317507.

Programs

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 *)

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.