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

%I A080092 #33 Feb 16 2025 08:32:48
%S A080092 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,
%T A080092 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,
%U A080092 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
%N A080092 Irregular triangle read by rows, giving prime sequences (p-1|2n) appearing in the n-th von Staudt-Clausen sum.
%C A080092 From _Gary W. Adamson_ & _Mats Granvik_, Aug 09 2008: (Start)
%C A080092 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.
%C A080092 A046886(n-1) = number of terms in row n.
%C A080092 The same terms in A143343 may be extracted from triangle A138239.
%C A080092 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.
%C A080092 The denominator of B_12 = 2730 = 2*3*5*7*13 = A027642(12) and A002445(6).
%C A080092 For example, B_12 = -691/2730 = (1 - 1/2 - 1/3 - 1/5 - 1/7 - 1/13).
%C A080092 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)
%H A080092 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/vonStaudt-ClausenTheorem.html">von Staudt-Clausen Theorem</a>.
%H A080092 Wikipedia, <a href="http://en.wikipedia.org/wiki/Von_Staudt-Clausen_theorem">Von Staudt-Clausen theorem</a>.
%e A080092 First few rows of the triangle:
%e A080092   2;
%e A080092   2, 3;
%e A080092   2, 3, 5;
%e A080092   2, 3, 7;
%e A080092   2, 3, 5;
%e A080092   2, 3, 11;
%e A080092   2, 3, 5, 7, 13;
%e A080092   2, 3;
%e A080092   ...
%e A080092 Sum for n=1 is 1/2 + 1/3, so terms are 2, 3;
%e A080092 sum for n=2 is 1/2 + 1/3 + 1/5, so terms are 2, 3, 5; etc.
%t A080092 row[n_] := Select[ Prime /@ Range[n+1], Divisible[2n, # - 1] &]; Flatten[Table[row[n], {n, 0, 25}]] (* _Jean-François Alcover_, Oct 12 2011 *)
%Y A080092 Cf. A000146, A002445, A027642, A138239, A143343.
%K A080092 nonn,easy,nice,tabf
%O A080092 1,1
%A A080092 _Eric W. Weisstein_, Jan 27 2003
%E A080092 Edited by _N. J. A. Sloane_, Nov 01 2009 at the suggestion of _R. J. Mathar_