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.

3, 6, 13, 27, 54, 109, 219, 439, 879, 1759, 3518, 7037, 14075, 28151, 56303, 112606, 225212, 450424, 900848, 1801696, 3603393, 7206787, 14413574, 28827148, 57654296, 115308593, 230617186, 461234373

Offset: 2

Author

Jianqiang Zhao (jqz(AT)math.upenn.edu), Jan 06 2003

Keywords

• more
• nonn

Comments

The n-th partial sum of double harmonic series is defined to be Sum_{1 <= k < l <= n} 1/(kl).

Examples

			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.
		

Links

• J. Zhao, Partial sums of multiple zeta value series II: finiteness of p-divisible sets, arXiv:math/0303043 [math.NT], 2003-2010. See (23) p. 11.

Crossrefs

Cf. A079404.

Programs

Maple

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:

Mathematica

nmax = 15; dhs = Array[HarmonicNumber[# - 1]/# &, 2^nmax] // Accumulate; Print["dhs finished"];
f[s_] := IntegerExponent[s // Denominator, 2];
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]]];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 2, nmax}] (* Jean-François Alcover, Jan 22 2018 *)

Formula

From Benoit Cloitre, Jan 24 2003: (Start)

a(n+1) - 2*a(n) = (a(n+1) mod 2);

a(n) = floor(c*2^n) where c = 1.718232... = 3/2 + Sum_{k>=2} (a(k+1) - 2*a(k))/2^k. (End)

Extensions

a(23)-a(29) from Sean A. Irvine, Aug 12 2025