1,1

Comments from Gary W. Adamson & Mats O. Granvik (qntmpkt(AT)yahoo.com), Aug 09 2008 (Start) The Von Staudt-Clausen theorem has two parts: generating denominators of the B_2n and the actual values. Both operations can be demonstrated in triangles A143343 and A080092 by following the procedures outlined in [Wikipedia - Bernoulli numbers] and summarized in A143343.

A143345 = number of terms in each row of A080092 (1, 2, 3, 3, 3, 3, 5, 2,...)

The same terms in A143343 may be extracted from triangle A138239.

Extract primes from even numbered rows of triangle A143343 but also include "2" as row 1. The rows are thus 1, 2, 4, 6...; generating denominators of B_1, B_2, B_4,...as well as B_1, B_2, B_4,...; as two parts of the Von Staudt-Clausen theorem.

The denominator of B_12 = 2730 = (2*3*5*7*13) = A027642(12) and A002445(6).

For example, B_12 = -691/2730 = (1 - 1/2 - 1/3 - 1/5 - 1/7 - 1/13)

The second operation is the Von Staudt-Clausen representation of Bn, obtained by starting with "1" then subtracting the reciprocals of terms in each row. (Cf. A143343 for a detailed explanation of the operations). (End)

Wikipedia (Bernoulli numbers).

Eric Weisstein's World of Mathematics, von Staudt-Clausen Theorem

First few rows of the triangle are:

2;

2, 3;

2, 3, 5;

2, 3, 7;

2, 3, 5;

2, 3, 11;

2, 3, 5, 7, 13;

2, 3;

...

Sum for n=1 is 1/2+1/3, so terms are 2, 3. Sum for n=2 is 1/2+1/3+1/5, so terms are 2, 3, 5. Etc.

Cf. A000146.

Cf. A000146, A143343, A143345, A138239, A002445, A027642.

Sequence in context: A079542 A022467 A037126 this_sequence A164738 A126225 A056160

Adjacent sequences: A080089 A080090 A080091 this_sequence A080093 A080094 A080095

nonn,easy,nice,tabf,new

Eric Weisstein (eric(AT)weisstein.com), Jan 27, 2003

Edited by njas, Nov 01 2009 at the suggestion of R. J. Mathar