

A165908


Irregular triangle with the terms in the StaudtClausen 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
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OFFSET

0,2


COMMENTS

The decomposition of a nonzero Bernoulli number in the StaudtClausen 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 = 11/21/31/11. Multiplied by the product 2*3*11=66 of the denominators this becomes 5=6633226, 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 (p1) = 0 then Ld := [op(Ld), 1/p] ; pp := pp*p ; elif p1 > 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}]]] (* JeanFrançois 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



