A395618 - OEIS

A395618

Decimal expansion of Sum_{k>=1} H(k)/((k+1)^2*2^k), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

%I #18 Jul 08 2026 14:55:00

%S 1,8,9,5,0,6,0,0,8,4,6,0,2,5,5,4,1,1,4,4,3,6,5,0,0,1,2,8,4,0,6,1,8,9,

%T 8,2,9,7,2,5,3,7,7,4,4,6,3,5,5,8,9,1,6,9,6,5,0,9,9,7,2,7,4,4,9,4,1,7,

%U 5,3,7,6,5,7,8,5,5,4,5,9,7,9,2,7,7,6,2,0,0,9,6,2,0,2,5,8,9,4,5,9,0,6,5,2,7

%N Decimal expansion of Sum_{k>=1} H(k)/((k+1)^2*2^k), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

%H Michael I. Shamos, <a href="https://web.archive.org/web/20220206102634/https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.366.9997&rep=rep1&type=pdf">Shamos's Catalog of the Real Numbers</a>, 2011.

%H Cornel Ioan Vălean, <a href="https://doi.org/10.1007/978-3-030-02462-8">(Almost) Impossible Integrals, Sums, and Series</a>, Springer International Publishing, 2019, section 4.51, p. 308, eq. (4.79), section 5.51, p. 326, section 6.51, pp. 498-502.

%H <a href="/index/Ha#harmonic">Index entries for sequences related to harmonic numbers</a>.

%F Equals zeta(3)/4 - log(2)^3/3.

%F Equals Integral_{x=0..1} log(1+x)^2/(x*(x+1)) dx = Integral_{x=1..2} log(x)^2/(x*(x-1)) dx (Shamos, 2011, p. 253).

%e 0.189506008460255411443650012840618982972537744635589...

%t RealDigits[Zeta[3]/4 - Log[2]^3/3, 10, 120][[1]]

%o (PARI) zeta(3)/4 - log(2)^3/3

%Y Cf. A001008, A002805.

%Y Cf. A002117, A002162.

%Y Sum_{k>=1} H(k)/((k+1)^m*2^k): A016627 (m=0), A253191 (m=1), this constant (m=2), A396733 (m=3), A396735 (m=4).

%K nonn,cons

%O 0,2

%A _Amiram Eldar_, Jun 04 2026