A195506 Denominator of Sum_{k=1..n} H(k)/k^2, where H(k) is the k-th harmonic number.

1, 8, 216, 1728, 216000, 216000, 74088000, 592704000, 16003008000, 16003008000, 21300003648000, 21300003648000, 46796108014656000, 46796108014656000, 46796108014656000, 374368864117248000, 1839274229408039424000, 1839274229408039424000

Offset: 1

Author

Franz Vrabec, Sep 19 2011

Keywords

• nonn
• frac
• easy

Comments

Lim_{n -> infinity} (A195505(n)/a(n)) = 2*Zeta(3) [L. Euler].

For n = 1 to n = 13, a(n) = A334582(n), but a(14) = 46796108014656000 <> 6685158287808000 = A334582(14). - Petros Hadjicostas, May 06 2020

Examples

			a(2) = 8 because 1 + (1 + 1/2)/2^2 = 11/8.
The first few fractions are 1, 11/8, 341/216, 2953/1728, 388853/216000, 403553/216000, 142339079/74088000, 1163882707/592704000, ... = A195505/A195506. - _Petros Hadjicostas_, May 06 2020
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..768
• Leonhard Euler, Meditationes circa singulare serierum genus, Novi. Comm. Acad. Sci. Petropolitanae, 20 (1775), 140-186.

Crossrefs

Cf. A002117, A195505 (numerators), A334582.

Programs

Mathematica

s = 0; Table[s = s + HarmonicNumber[n]/n^2; Denominator[s], {n, 20}] (* T. D. Noe, Sep 20 2011 *)

PARI

H(n) = sum(k=1, n, 1/k);
a(n) = denominator(sum(k=1, n, H(k)/k^2)); \\ Michel Marcus, May 07 2020