A080092 Irregular triangle read by rows, giving prime sequences (p-1|2n) appearing in the n-th von Staudt-Clausen sum.

2, 2, 3, 2, 3, 5, 2, 3, 7, 2, 3, 5, 2, 3, 11, 2, 3, 5, 7, 13, 2, 3, 2, 3, 5, 17, 2, 3, 7, 19, 2, 3, 5, 11, 2, 3, 23, 2, 3, 5, 7, 13, 2, 3, 2, 3, 5, 29, 2, 3, 7, 11, 31, 2, 3, 5, 17, 2, 3, 2, 3, 5, 7, 13, 19, 37, 2, 3, 2, 3, 5, 11, 41, 2, 3, 7, 43, 2, 3, 5, 23, 2, 3, 47, 2, 3, 5, 7, 13, 17, 2, 3

Offset

1,1

Comments

From Gary W. Adamson & Mats Granvik, 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.

A046886(n-1) = number of terms in row n.

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" and then subtracting the reciprocals of terms in each row. (Cf. A143343 for a detailed explanation of the operations.) (End)

Links

Table of n, a(n) for n=1..92.

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

Wikipedia, Von Staudt-Clausen theorem.

Example

First few rows of the triangle:
  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.

Mathematica

row[n_] := Select[ Prime /@ Range[n+1], Divisible[2n, # - 1] &]; Flatten[Table[row[n], {n, 0, 25}]] (* Jean-François Alcover, Oct 12 2011 *)

Crossrefs

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

Sequence in context: A306894 A169614 A037126 * A164738 A126225 A306997

Adjacent sequences: A080089 A080090 A080091 * A080093 A080094 A080095

Keyword

nonn,easy,nice,tabf

Author

Eric W. Weisstein, Jan 27 2003

Extensions

Edited by N. J. A. Sloane, Nov 01 2009 at the suggestion of R. J. Mathar

Status

approved