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%I #34 May 10 2020 13:42:31
%S 1,8,216,1728,216000,216000,74088000,592704000,16003008000,
%T 16003008000,21300003648000,21300003648000,46796108014656000,
%U 46796108014656000,46796108014656000,374368864117248000,1839274229408039424000,1839274229408039424000
%N Denominator of Sum_{k=1..n} H(k)/k^2, where H(k) is the k-th harmonic number.
%C Lim_{n -> infinity} (A195505(n)/a(n)) = 2*Zeta(3) [L. Euler].
%C For n = 1 to n = 13, a(n) = A334582(n), but a(14) = 46796108014656000 <> 6685158287808000 = A334582(14). - _Petros Hadjicostas_, May 06 2020
%H Seiichi Manyama, <a href="/A195506/b195506.txt">Table of n, a(n) for n = 1..768</a>
%H Leonhard Euler, <a href="http://eulerarchive.maa.org/pages/E477.html">Meditationes circa singulare serierum genus</a>, Novi. Comm. Acad. Sci. Petropolitanae, 20 (1775), 140-186.
%e a(2) = 8 because 1 + (1 + 1/2)/2^2 = 11/8.
%e 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
%t s = 0; Table[s = s + HarmonicNumber[n]/n^2; Denominator[s], {n, 20}] (* _T. D. Noe_, Sep 20 2011 *)
%o (PARI) H(n) = sum(k=1, n, 1/k);
%o a(n) = denominator(sum(k=1, n, H(k)/k^2)); \\ _Michel Marcus_, May 07 2020
%Y Cf. A002117, A195505 (numerators), A334582.
%K nonn,frac,easy
%O 1,2
%A _Franz Vrabec_, Sep 19 2011