A157250 Wilson numbers: k such that the generalized Wilson quotient A157249(k) is divisible by k.

1, 5, 13, 563, 5971, 558771, 1964215, 8121909, 12326713, 23025711, 26921605, 341569806, 399292158

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

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, 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.

Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.

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.

Mathematica

f[n_] := Times @@ Select[Range[n], CoprimeQ[n, #]&];
e[1|2|4] = 1; e[n_] := If[MatchQ[FactorInteger[n], {{_?OddQ, _}} | {{2, 1}, {_, _}}], 1, -1];
WilsonQ[n_] := IntegerQ[(f[n] + e[n])/n^2];
Reap[For[k = 1, k < 10^7, k++, If[WilsonQ[k], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Dec 11 2018 *)

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: A012033 A007540 A290171 * A009157 A153374 A247789

Adjacent sequences: A157247 A157248 A157249 * A157251 A157252 A157253

Keyword

more,nonn

Author

Jonathan Sondow and Wadim Zudilin, Feb 27 2009

Status

approved