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A079404
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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).
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1
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0, 1, 1, 3, 4, 3, 3, 5, 7, 9, 10, 9, 10, 12, 14, 13, 13, 15, 17, 19, 19
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OFFSET
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2,4
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COMMENTS
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The sequence is conjectured to go to positive infinity.
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REFERENCES
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Partial sums of multiple zeta value series II: finiteness of p-divisible sets.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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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;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Jianqiang Zhao (jqz(AT)math.upenn.edu), Jan 06 2003
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EXTENSIONS
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STATUS
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approved
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