Wolstenholme Number

The Wolstenholme numbers are defined as the numerators of the generalized harmonic number H_(n,2) appearing in Wolstenholme's theorem. The first few are 1, 5, 49, 205, 5269, 5369, 266681, 1077749, ... (OEIS A007406).

By Wolstenholme's theorem, p|a_(p-1) for prime p>3, where a_n is the nth Wolstenholme number. In addition, p|a_((p-1)/2) for prime p>3.

The first few prime Wolstenholme numbers are 5, 266681, 40799043101, 86364397717734821, ... (OEIS A123751), which occur at indices n=2, 7, 13, 19, 121, 188, 252, 368, 605, 745, ... (OEIS A111354).

See also

Harmonic Number, Integer Sequence Primes, Wolstenholme Prime, Wolstenholme's Theorem

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Savio, D. Y.; Lamagna, E. A.; and Liu, S.-M. "Summation of Harmonic Numbers." In Computers and Mathematics (Ed. E. Kaltofen and S. M. Watt). New York: Springer-Verlag, pp. 12-20, 1989.Sloane, N. J. A. Sequences A007406/M4004, A111354, and A123751 in "The On-Line Encyclopedia of Integer Sequences."

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Wolstenholme Number

Cite this as:

Weisstein, Eric W. "Wolstenholme Number." From MathWorld--A Wolfram Web Resource.

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