cp's OEIS Frontend

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

%I A165908 #15 May 05 2015 12:50:01
%S A165908 1,2,-1,6,-3,-2,30,-15,-10,-6,42,-21,-14,-6,30,-15,-10,-6,66,-33,-22,
%T A165908 -6,2730,-1365,-910,-546,-390,-210,12,-3,-2,-3060,-255,-170,-102,-30,
%U A165908 44688,-399,-266,-114,-42
%N A165908 Irregular triangle with the terms in the Staudt-Clausen theorem for the nonzero Bernoulli numbers multiplied by the product of the associated primes.
%C A165908 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.
%C A165908 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 .
%H A165908 R. Mestrovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mestrovic/mes4.html">On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers</a>, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
%e A165908 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.
%e A165908 1;
%e A165908 2,-1;
%e A165908 6,-3,-2;
%e A165908 30,-15,-10,-6;
%e A165908 42,-21,-14,-6;
%e A165908 30,-15,-10,-6;
%e A165908 66,-33,-22,-6;
%e A165908 2730,-1365,-910,-546,-390,-210;
%p A165908 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
%t A165908 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 *)
%Y A165908 Cf. A000146, A165884, A006954 (first column).
%K A165908 tabf,sign
%O A165908 0,2
%A A165908 _Paul Curtz_, Sep 30 2009
%E A165908 Edited by _R. J. Mathar_, Jul 08 2011