OFFSET
2,4
COMMENTS
The sequence is conjectured to go to positive infinity.
REFERENCES
Partial sums of multiple zeta value series II: finiteness of p-divisible sets.
LINKS
J. Zhao, Partial sums of multiple zeta value series II: finiteness of p-divisible sets, arXiv:math/0303043 [math.NT], 2003-2010. See (24) p. 11.
EXAMPLE
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.
MAPLE
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:
MATHEMATICA
nmax = 15; dhs = Array[HarmonicNumber[# - 1 ]/# &, 2^nmax] // Accumulate; Print["dhs finished"];
f[s_] := IntegerExponent[s // Denominator, 2];
a[n_] := Table[{f[dhs[[k]] ], k}, {k, 2^(n - 1), 2^n - 1}] // Sort // First // First;
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 2, nmax}] (* Jean-François Alcover, Jan 22 2018 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Jianqiang Zhao (jqz(AT)math.upenn.edu), Jan 06 2003
EXTENSIONS
Typo in data corrected by Jean-François Alcover, Jan 22 2018
STATUS
approved