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

1,2

Links

Robert Israel, Table of n, a(n) for n = 1..1000

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).

Example

The squares of the first three harmonic numbers are 1, 9/4, 121/36 which sum to 119/18 so a(3)=119.

Maple

H2:= n-> add(harmonic(k)^2, k = 1..n): seq(numer(H2(n)), n=1..25);

Mathematica

Accumulate[HarmonicNumber[Range[30]]^2]//Numerator (* Harvey P. Dale, Aug 10 2025 *)

Prog

(PARI) a(n) = numerator(sum(k=1, n, sum(i=1, k, 1/i)^2)); \\ Michel Marcus, Apr 07 2025

Crossrefs

Cf. A001008, A002805, A382813 (denominators).

Cf. A027611, A027612

Sequence in context: A367244 A016285 A121086 * A159969 A253512 A295048

Adjacent sequences: A382809 A382810 A382811 * A382813 A382814 A382815

Keyword

nonn,frac

Author

Gary Detlefs, Apr 05 2025

Status

approved