von Staudt-Clausen Theorem

$$B_{2n} = A_n - \sum_{\scriptstyle p_k\atop\scriptstyle p_k-1\vert 2n} {1\over p_k},$$

where $B_{2n}$ is a Bernoulli Number, $A_n$ is an Integer, and the $p_k$s are the Primes satisfying $p_k-1\vert 2k$. For example, for $k=1$, the primes included in the sum are 2 and 3, since $(2-1)\vert 2$ and $(3-1)\vert 2$. Similarly, for $k=6$, the included primes are (2, 3, 5, 7, 13), since (1, 2, 3, 6, 12) divide $12=2\cdot 6$. The first few values of $A_n$ for $n=1$, 2, ... are 1, 1, 1, 1, 1, 1, 2, $-6$, 56, $-528$, ... (Sloane's A000146).

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

See also Bernoulli Number, 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.