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A165908 Irregular triangle with the terms in the Staudt-Clausen theorem for the nonzero Bernoulli numbers multiplied by the product of the associated primes. 3
1, 2, -1, 6, -3, -2, 30, -15, -10, -6, 42, -21, -14, -6, 30, -15, -10, -6, 66, -33, -22, -6, 2730, -1365, -910, -546, -390, -210, 12, -3, -2, -3060, -255, -170, -102, -30, 44688, -399, -266, -114, -42 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The decomposition of a nonzero Bernoulli number in the Staudt-Clausen format is B(n) = A000146(n) - sum_k 1/A080092(n,k) with a set of primes A080092 characterizing the right hand side.

If we multiply this equation by the product of the primes for a given n (which is in A002445), discard the left hand side, and list individually the terms associated with A000146 and each of the k, we get row n of the current triangle .

LINKS

Table of n, a(n) for n=0..40.

R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.

EXAMPLE

The decomposition of B_10 is 5/66 = 1-1/2-1/3-1/11. Multiplied by the product 2*3*11=66 of the denominators this becomes 5=66-33-22-6, and the 4 terms on the right hand side become one row of the table.

1;

2,-1;

6,-3,-2;

30,-15,-10,-6;

42,-21,-14,-6;

30,-15,-10,-6;

66,-33,-22,-6;

2730,-1365,-910,-546,-390,-210;

MAPLE

A165908 := proc(n) local i, p; Ld := [] ; pp := 1 ; for i from 1 do p := ithprime(i) ; if (2*n) mod (p-1) = 0 then Ld := [op(Ld), -1/p] ; pp := pp*p ; elif p-1 > 2*n then break; end if; end do: Ld := [A000146(n), op(Ld)] ; [seq(op(i, Ld)*pp, i=1..nops(Ld))] ; end proc: # for n>=2, R. J. Mathar, Jul 08 2011

MATHEMATICA

a146[n_] := Sum[ Boole[ PrimeQ[d+1]]/(d+1), {d, Divisors[2n]}] + BernoulliB[2n]; primes[n_] := Select[ Prime /@ Range[n+1], Divisible[2n, #-1]& ]; row[n_] := With[{pp = primes[n]}, Join[{a146[n]}, -1/pp]*Times @@ pp]; Join[{1}, Flatten[ Table[row[n], {n, 0, 9}]]] (* Jean-Fran├žois Alcover_, Aug 09 2012 *)

CROSSREFS

Cf. A000146, A165884, A006954 (first column).

Sequence in context: A094307 A097905 A094310 * A121281 A232467 A131449

Adjacent sequences:  A165905 A165906 A165907 * A165909 A165910 A165911

KEYWORD

tabf,sign

AUTHOR

Paul Curtz, Sep 30 2009

EXTENSIONS

Edited by R. J. Mathar, Jul 08 2011

STATUS

approved

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Last modified December 6 03:02 EST 2016. Contains 278771 sequences.