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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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: A341793 A341773 A096142 * A344982 A280285 A033719

Adjacent sequences:  A216918 A216919 A216920 * A216922 A216923 A216924

KEYWORD

nonn

AUTHOR

Peter Luschny, Sep 20 2012

STATUS

approved

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Last modified July 23 14:44 EDT 2021. Contains 346259 sequences. (Running on oeis4.)