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A157249
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Generalized Wilson quotients (or Wilson quotients for composite moduli).
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5
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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
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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L. E. Dickson, History of the Theory of Numbers, vol. 1, Divisibility and Primality, Chelsea, New York, 1966.
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LINKS
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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.
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FORMULA
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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.
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EXAMPLE
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P(8) = 3*5*7 = 105 and e(8) = -1, so a(8) = (105-1)/8 = 13.
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MAPLE
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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:
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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