cp's OEIS Frontend

A079403 Let G(t) be the set of numbers between 2^(t-1) and 2^t-1, inclusive. There is a unique number a(t) in G(t) so that the denominator of the a(t)-th partial sum of the double harmonic series is divisible by smaller 2-powers than its neighbors.

%I A079403 #24 Aug 12 2025 20:31:16
%S A079403 3,6,13,27,54,109,219,439,879,1759,3518,7037,14075,28151,56303,112606,
%T A079403 225212,450424,900848,1801696,3603393,7206787,14413574,28827148,
%U A079403 57654296,115308593,230617186,461234373
%N A079403 Let G(t) be the set of numbers between 2^(t-1) and 2^t-1, inclusive. There is a unique number a(t) in G(t) so that the denominator of the a(t)-th partial sum of the double harmonic series is divisible by smaller 2-powers than its neighbors.
%C A079403 The n-th partial sum of double harmonic series is defined to be Sum_{1 <= k < l <= n} 1/(kl).
%H A079403 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 (23) p. 11.
%F A079403 From _Benoit Cloitre_, Jan 24 2003: (Start)
%F A079403 a(n+1) - 2*a(n) = (a(n+1) mod 2);
%F A079403 a(n) = floor(c*2^n) where c = 1.718232... = 3/2 + Sum_{k>=2} (a(k+1) - 2*a(k))/2^k. (End)
%e A079403 a(3)=6 because Sum_{1 <= k < l <= 6} 1/(kl) = 203/90, 4 does not divide 90, while 4 divides the denominators of both Sum_{1 <= k < l <= 5} 1/(kl) = 15/8 and Sum_{1 <= k < l <= 7} 1/(kl) = 469/180.
%p A079403 sequ := proc(T) local A,i,n,t,psum,innersum; psum := 0; innersum := 0; A := {}; for t to T-1 do for n from 2^t to 2^(t+1)-1 do innersum := innersum+2^T/(n-1) mod 2^(2*T); psum := psum+2^T*innersum/n mod 2^(2*T); if psum mod 2^(2*T-t+1)=0 then A := A union {n}; end if; od; od; RETURN(A); end:
%t A079403 nmax = 15; dhs = Array[HarmonicNumber[# - 1]/# &, 2^nmax] // Accumulate; Print["dhs finished"];
%t A079403 f[s_] := IntegerExponent[s // Denominator, 2];
%t A079403 a[2] = 3; a[n_] := a[n] = For[k = 2*a[n - 1], k <= 2^n - 1, k++, fk = f[dhs[[k]]]; If[f[dhs[[k-1]]] > fk && f[dhs[[k+1]]] > fk, Return[k]]];
%t A079403 Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 2, nmax}] (* _Jean-François Alcover_, Jan 22 2018 *)
%Y A079403 Cf. A079404.
%K A079403 more,nonn
%O A079403 2,1
%A A079403 Jianqiang Zhao (jqz(AT)math.upenn.edu), Jan 06 2003
%E A079403 a(23)-a(29) from _Sean A. Irvine_, Aug 12 2025