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.
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.
is a Bernoulli number, 
is an integer, and the 
s
are the primes satisfying 
, i.e., 
divides 
.
, the primes included in the sum are 2 and 3, since 
and 
,
giving
, the included primes are (2, 3, 5, 7, 13), since (1, 2,
4, 6, 12) divide 
, giving
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).
