von Staudt-Clausen Theorem

where is a Bernoulli Number, is an Integer, and the s are the Primes satisfying . For example, for , the primes included in the sum are 2 and 3, since and . Similarly, for , the included primes are (2, 3, 5, 7, 13), since (1, 2, 3, 6, 12) divide . The first few values of for , 2, ... are 1, 1, 1, 1, 1, 1, 2, , 56, , ... (Sloane's A000146).

The theorem was rediscovered by Ramanujan (Hardy 1959, p. 11) and can be proved using p-adic Number.

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 109, 1996.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1959.

Hardy, G. H. and Wright, E. M. The Theorem of von Staudt'' and Proof of von Staudt's Theorem.'' §7.9-7.10 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 90-93, 1979.

Sloane, N. J. A. Sequence A000146/M1717 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Staudt, K. G. C. von. Beweis eines Lehrsatzes, die Bernoullischen Zahlen betreffend.'' J. reine angew. Math. 21, 372-374, 1840.