A165908 Irregular triangle with the terms in the Staudt-Clausen theorem for the nonzero Bernoulli numbers multiplied by the product of the associated primes.

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

Offset: 0

Author

Paul Curtz, Sep 30 2009

Keywords

• tabf
• sign

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 .

Examples

			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;
		

Links

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

Crossrefs

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

Programs

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-François Alcover_, Aug 09 2012 *)

Extensions

Edited by R. J. Mathar, Jul 08 2011