A216921 a(n) = 2*n+1 - gpf(denominator(B°(2*n))) where B°(n) are Zagier's modification of the Bernoulli numbers and gpf(n) is the greatest prime factor of n.

0, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 2, 24, 0, 0, 2, 32, 0, 2, 0, 0, 2, 0, 2, 40, 0, 2, 28, 0, 0, 2, 34, 0, 2, 0, 0, 2, 40, 0, 2, 0, 2, 84, 0, 2, 46, 92, 0, 2, 0, 0, 2, 0, 0, 2, 0, 2, 58, 116, 60, 120, 64, 0, 2, 0, 2, 132, 0, 0, 2, 140, 72, 144, 0, 0, 2, 132, 0

Offset

1,4

Comments

Dixit and others (see link, p.13) wrote: "The data suggests that the prime factors of alpha(2n) [= denominator(B°(2n))] are bounded by 2n + 1." If this is true a(n) will never become negative. The data also suggests that a(n) = 0 only if 2n+1 is prime.

Links

Table of n, a(n) for n=1..78.

A. Dixit, V. H. Moll, Ch. Vignat, The Zagier modification of Bernoulli numbers and a polynomial extension. Part I, arXiv:1209.4110v1 [math.NT], 2010.

D. Zagier. A modified Bernoulli number, Nieuw Archief voor Wiskunde, 16:63-72, 1998.

Formula

a(n) = 2*n+1 - A006530(A216923(2*n)).

Maple

A216921 := proc(n) local zb2, F;
zb2 := denom(add(binomial(2*n+r, 2*r)*bernoulli(r)/(2*n+r), r=0..2*n));
F := ifactors(zb2)[2]; 2*n+1-F[nops(F)][1] end;

Mathematica

b[n_] := Sum[Binomial[n + k, 2*k]*BernoulliB[k]/(n + k), {k, 0, n}] // Denominator;
a[n_] := 2*n + 1 - FactorInteger[b[2*n]][[-1, 1]];
Array[a, 80] (* Jean-François Alcover, Nov 29 2017 *)

Prog

(PARI) f(n) = denominator(sum(r=0, n, binomial(n+r, 2*r)*bernfrac(r)/(n+r))); \\ A216923
a(n) = 2*n+1 - vecmax(factor(f(2*n))[, 1]); \\ Michel Marcus, Sep 29 2019

Crossrefs

Cf. A006530, A216923.

Sequence in context: A341773 A359288 A096142 * A344982 A359240 A280285

Adjacent sequences: A216918 A216919 A216920 * A216922 A216923 A216924

Keyword

nonn

Author

Peter Luschny, Sep 20 2012

Status

approved