von Staudt-Clausen Theorem

The von Staudt-Clausen theorem, sometimes also known as the Staudt-Clausen theorem (Carlitz 1968), states that

(1)

where  is a Bernoulli number,  is an integer, and the s are the primes satisfying , i.e.,  divides .

For example, for , the primes included in the sum are 2 and 3, since  and , giving

(2)

Similarly, for , the included primes are (2, 3, 5, 7, 13), since (1, 2, 4, 6, 12) divide , giving

(3)

The first few values of  for , 2, ... are 1, 1, 1, 1, 1, 1, 2, , 56, , ... (OEIS A000146), and the lists of primes appearing in successive sums are 2, 3; 2, 3, 5; 2, 3, 7; 2, 3, 5; 2, 3, 11; ... (OEIS A080092).

The theorem was rediscovered by Ramanujan (Hardy 1999, p. 11) and can be proved using p-adic Numbers.

REFERENCES:

Carlitz, L. "Bernoulli Numbers." Fib. Quart. 6, 71-85, 1968.

Clausen, T. "Theorem." Astron. Nach. 17, 351-352, 1840.

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, 1999.

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.

Rado, R. "A New Proof of a Theorem of V. Staudt." J. London Math. Soc. 9, 85-88, 1934.

Rado, R. "A Note on the Bernoullian Numbers." J. London Math. Soc. 9, 88-90, 1934.

Sloane, N. J. A. Sequences A000146/M1717 and A080092 in "The On-Line Encyclopedia of Integer Sequences."

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