1,2

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.

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.

A Wilson number is prime if and only if it is a Wilson prime A007540. Only three are known: 5, 13, 563.

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.

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, p. 65.

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

A157249(n) == 0 mod n.

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.

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.

Cf. Wilson quotient A007619, Wilson prime A007540, generalized Wilson quotient A157249, n-phi-torial A001783, numbers having a primitive root A033948.

Sequence in context: A145557 A012033 A007540 this_sequence A155185 A009157 A153374

Adjacent sequences: A157247 A157248 A157249 this_sequence A157251 A157252 A157253

nonn

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