A397029 Decimal expansion of Sum_{k>=1} H(k) / (2*k+1)^3, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

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

Offset

0,2

Links

Table of n, a(n) for n=0..104.

Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2nd ed., 2023, section 4.1.14, pp. 273-275.

Cornel Ioan Vălean, More (Almost) Impossible Integrals, Sums, and Series, Springer Cham, 2023. See section 4.19, pp. 419-420, eq.(4.101), section 5.19, pp. 460-461, and section 6.19, pp. 595-601.

Index entries for sequences related to harmonic numbers.

Formula

Equals Pi^4/64 - 7*zeta(3)*log(2)/4 = 45*zeta(4)/32 - 7*zeta(3)*log(2)/4.

Example

0.063912929135327093289295771353287181461905748354964...

Mathematica

RealDigits[45*Zeta[4]/32 - 7*Zeta[3]*Log[2]/4, 10, 120, -1][[1]]

Prog

(PARI) 45*zeta(4)/32 - 7*zeta(3)*log(2)/4

Crossrefs

Cf. A001008, A002805.

Cf. A002117, A002162, A013662.

Sum_{k>=1} H(k) / (2*k+1)^m: A397028 (m=2), this constant (m=3), A397030 (m=4), A397031 (m=5), A397032 (m=6).

Sequence in context: A242962 A384585 A257938 * A153632 A308170 A197511

Adjacent sequences: A397026 A397027 A397028 * A397030 A397031 A397032

Keyword

nonn,cons

Author

Amiram Eldar, Jun 14 2026

Status

approved