Von Staudt–Clausen theorem

In number theory, the von Staudt–Clausen theorem is a result determining the fractional part of Bernoulli numbers, found independently by Karl von Staudt (1840) and Thomas Clausen (1840).

Specifically, if n is a positive integer and we add 1/p to the Bernoulli number B_2n for every prime p such that p − 1 divides 2n, then we obtain an integer; that is,

${\displaystyle B_{2n}+\sum _{(p-1)|2n}{\frac {1}{p}}\in \mathbb {Z} .}$

This fact immediately allows us to characterize the denominators of the non-zero Bernoulli numbers B_2n as the product of all primes p such that p − 1 divides 2n; consequently, the denominators are square-free and divisible by 6.

These denominators are

6, 30, 42, 30, 66, 2730, 6, 510, 798, 330, 138, 2730, 6, 870, 14322, 510, 6, 1919190, 6, 13530, ... (sequence A002445 in the OEIS).

The sequence of integers ${\displaystyle B_{2n}+\sum _{(p-1)|2n}{\frac {1}{p}}}$ is

1, 1, 1, 1, 1, 1, 2, -6, 56, -528, 6193, -86579, 1425518, -27298230, ... (sequence A000146 in the OEIS).