Search: id:A157250
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%I A157250
%S A157250 1,5,13,563,5971,558771,1964215,8121909,12326713,23025711,
%T A157250 26921605,341569806,399292158
%N A157250 Wilson numbers: n such that the generalized Wilson quotient A157249(n) 
               is divisible by n.
%C A157250 A prime p is a Wilson prime if p divides its Wilson quotient A007619. 
               A number n is a Wilson number if n divides its generalized Wilson 
               quotient A157249.
%C A157250 The sequence contains all Wilson numbers <= 5 x 10^8. Heuristics suggest 
               that #(Wilson numbers < N) is about (6/pi^2) log N, for large N.
%C A157250 A Wilson number is prime if and only if it is a Wilson prime A007540. 
               Only three are known: 5, 13, 563.
%C A157250 The first composite Wilson number 5971 was discovered by Kloss, the others 
               by Agoh, Dilcher, and Skula. Every known composite Wilson number 
               n has at least two odd prime factors, so e(n) = -1.
%C A157250 For additional references and links, see A007540.
%D A157250 T. Agoh, K. Dilcher, and L. Skula, Wilson quotients for composite moduli, 
               Math. Comp. 67 (1998), 843-861.
%D A157250 K. E. Kloss, Some number theoretic calculations, J. Res. Nat. Bureau 
               of Stand., B, 69 (1965), 335-339.
%D A157250 L. E. Dickson, History of the Theory of Numbers, vol. 1, Divisibility 
               and Primality, Chelsea, New York, 1966, p. 65.
%H A157250 T. Agoh, K. Dilcher, and L. Skula, <a href="http://www.ams.org/mcom/1998-67-222/
               S0025-5718-98-00951-X/home.html"> Wilson quotients for composite 
               moduli</a>.
%F A157250 A157249(n) == 0 mod n.
%F A157250 A001783(n) + e(n) == 0 mod n^2, where e(n) = +1 or -1 according as n 
               does or does not have a primitive root.
%e A157250 A157249(13) = (A001783(13) + e(13))/13 = ((13-1)! + 1)/13 = 479001601/
               13 = 36846277 == 0 mod 13, so 13 is a member. A001783(5971) + e(5971) 
               = A001783(5971) - 1 == 0 mod 5971^2, so 5971 is a member. But A157249(8) 
               = (A001783(8) + e(8))/8 = (3*5*7 - 1)/8 = 13 ==/== 0 mod 8, so 8 
               is not a member.
%Y A157250 Cf. Wilson quotient A007619, Wilson prime A007540, generalized Wilson 
               quotient A157249, n-phi-torial A001783, numbers having a primitive 
               root A033948.
%Y A157250 Sequence in context: A145557 A012033 A007540 this_sequence A155185 A009157 
               A153374
%Y A157250 Adjacent sequences: A157247 A157248 A157249 this_sequence A157251 A157252 
               A157253
%K A157250 nonn
%O A157250 1,2
%A A157250 Jonathan Sondow (jsondow(AT)alumni.princeton.edu) and Wadim Zudilin (wzudilin(AT)mpim-bonn.mpg.de), 
               Feb 27 2009

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