A382812 Numerator of the n-th partial sum of the squares of the harmonic numbers.
1, 13, 119, 1577, 3233, 8867, 141563, 2844129, 28119709, 335676251, 3968696491, 55023970333, 758025067309, 799020611041, 1676892996083, 59597395635137, 351844709221043, 2314823924364859, 9114392136427625, 628176680098075, 216039223801697, 5117413095318143, 363066107054194281, 27957386425926920257
Offset: 1
Examples
The squares of the first three harmonic numbers are 1, 9/4, 121/36 which sum to 119/18 so a(3)=119.
Links
- Robert Israel, Table of n, a(n) for n = 1..1000
Programs
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Maple
H2:= n-> add(harmonic(k)^2, k = 1..n): seq(numer(H2(n)), n=1..25);
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Mathematica
Accumulate[HarmonicNumber[Range[30]]^2]//Numerator (* Harvey P. Dale, Aug 10 2025 *)
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PARI
a(n) = numerator(sum(k=1, n, sum(i=1, k, 1/i)^2)); \\ Michel Marcus, Apr 07 2025
Formula
a(n) = numerator((n+1)*H(n)^2-(2*n+1)*H(n) + 2*n), where H(n) is the n-th harmonic number.
a(n) = numerator((S(n)*H(n)^2 + (2*n - 2*S(n) + 1)*H(n)-2*n)/(H(n)-1)), where S(n) is the n-th partial sum of H(n).