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A157250 Wilson numbers: n such that the generalized Wilson quotient A157249(n) is divisible by n. 1
1, 5, 13, 563, 5971, 558771, 1964215, 8121909, 12326713, 23025711, 26921605, 341569806, 399292158 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

REFERENCES

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.

LINKS

Table of n, a(n) for n=1..13.

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

J. 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

FORMULA

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.

EXAMPLE

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.

CROSSREFS

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 * A155185 A009157 A153374

Adjacent sequences:  A157247 A157248 A157249 * A157251 A157252 A157253

KEYWORD

more,nonn

AUTHOR

Jonathan Sondow and Wadim Zudilin, Feb 27 2009

STATUS

approved

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Last modified July 29 14:58 EDT 2014. Contains 245039 sequences.