A216921 - OEIS

%I #24 Sep 29 2019 07:21:57

%S 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,

%T 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,

%U 64,0,2,0,2,132,0,0,2,140,72,144,0,0,2,132,0

%N 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.

%C 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.

%H A. Dixit, V. H. Moll, Ch. Vignat, <a href="http://arxiv.org/abs/1209.4110">The Zagier modification of Bernoulli numbers and a polynomial extension. Part I</a>, arXiv:1209.4110v1 [math.NT], 2010.

%H D. Zagier. <a href="http://people.mpim-bonn.mpg.de/zagier/files/tex/ModifiedBernoulliNum/fulltext.pdf">A modified Bernoulli number</a>, Nieuw Archief voor Wiskunde, 16:63-72, 1998.

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

%p A216921 := proc(n) local zb2, F;

%p zb2 := denom(add(binomial(2*n+r,2*r)*bernoulli(r)/(2*n+r),r=0..2*n));

%p F := ifactors(zb2)[2]; 2*n+1-F[nops(F)][1] end;

%t b[n_] := Sum[Binomial[n + k, 2*k]*BernoulliB[k]/(n + k), {k, 0, n}] // Denominator;

%t a[n_] := 2*n + 1 - FactorInteger[b[2*n]][[-1, 1]];

%t Array[a, 80] (* _Jean-François Alcover_, Nov 29 2017 *)

%o (PARI) f(n) = denominator(sum(r=0, n, binomial(n+r, 2*r)*bernfrac(r)/(n+r))); \\ A216923

%o a(n) = 2*n+1 - vecmax(factor(f(2*n))[,1]); \\ _Michel Marcus_, Sep 29 2019

%Y Cf. A006530, A216923.

%K nonn

%O 1,4

%A _Peter Luschny_, Sep 20 2012