cp's OEIS Frontend

A079404 Let G(n) be the set of numbers between 2^(n-1) and 2^n-1, inclusive. There is a unique number m(n) in G(n) so that the denominator of the m(n)-th partial sum of the double harmonic series is divisible by smaller 2-power than that of others in G(n). This power is defined to be a(n).

%I A079404 #20 Jan 22 2018 13:26:32
%S A079404 0,1,1,3,4,3,3,5,7,9,10,9,10,12,14,13,13,15,17,19,19
%N A079404 Let G(n) be the set of numbers between 2^(n-1) and 2^n-1, inclusive. There is a unique number m(n) in G(n) so that the denominator of the m(n)-th partial sum of the double harmonic series is divisible by smaller 2-power than that of others in G(n). This power is defined to be a(n).
%C A079404 The sequence is conjectured to go to positive infinity.
%D A079404 Partial sums of multiple zeta value series II: finiteness of p-divisible sets.
%H A079404 J. Zhao, <a href="https://arxiv.org/abs/math/0303043">Partial sums of multiple zeta value series II: finiteness of p-divisible sets</a>, arXiv:math/0303043 [math.NT], 2003-2010. See (24) p. 11.
%e A079404 a(3)=1 because G(3)={4,5,6,7} and among Sum_{1 <= k < l <= 4} 1/(kl) = 35/24, Sum_{1 <= k < l <= 5} 1/(kl) = 15/8, Sum_{1 <= k < l <= 6} 1/(kl) = 203/90, Sum_{1 <= k < l <= 7} 1/(kl) = 469/180, 90 has the smallest 2-power factor among the denominators.
%p A079404 sequ := proc(T) local b,counter,A,n,t,psum,innersum; psum := 0; innersum := 0; A := array(1..T-1); for t to T-1 do for n from 2^(t) to 2^(t+1)-1 do innersum := innersum+1/(n-1); psum := psum+innersum/n; if 2^(2*t)*psum mod 2^(2*t+1)=0 then print(`The conjecture that 2 never divides the numerators of partial sums of double harmonic series is wrong.`); else b := 0; counter := 2*t; while b=0 do b := 2^counter*psum mod 2; counter := counter-1; od; if counter<t-1 then A[t] := counter+1: end if; end if; od; od; RETURN(eval(A)): end:
%t A079404 nmax = 15; dhs = Array[HarmonicNumber[# - 1 ]/# &, 2^nmax] // Accumulate; Print["dhs finished"];
%t A079404 f[s_] := IntegerExponent[s // Denominator, 2];
%t A079404 a[n_] := Table[{f[dhs[[k]] ], k}, {k, 2^(n - 1), 2^n - 1}] // Sort // First // First;
%t A079404 Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 2, nmax}] (* _Jean-François Alcover_, Jan 22 2018 *)
%Y A079404 Cf. A079403.
%K A079404 nonn
%O A079404 2,4
%A A079404 Jianqiang Zhao (jqz(AT)math.upenn.edu), Jan 06 2003
%E A079404 Typo in data corrected by _Jean-François Alcover_, Jan 22 2018