A398076 Decimal expansion of Sum_{k>=1} (zeta(3) - H(2*k,3)) * H(k)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number, and H(k,3) = A007408(k)/A007409(k) is the k-th harmonic number of order 3.

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

Offset

0,3

Links

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

Cornel Ioan Vălean, More (Almost) Impossible Integrals, Sums, and Series, Springer Cham, 2023. See section 4.36, p. 435, eq. (4.131), section 5.36, p. 464, section 6.36, pp. 678-679.

Index entries for sequences related to harmonic numbers.

Formula

Equals 23*zeta(2)*zeta(3)/8 - 69*zeta(5)/16 + 7*log(2)^2*zeta(3)/4 - 45*log(2)*zeta(4)/16.

Example

0.113716171243622769812163652869964314620240807027934...

Mathematica

RealDigits[23*Zeta[2]*Zeta[3]/8 - 69*Zeta[5]/16 + 7*Log[2]^2*Zeta[3]/4 - 45*Log[2]*Zeta[4]/16, 10, 120][[1]]

Prog

(PARI) 23*zeta(2)*zeta(3)/8 - 69*zeta(5)/16 + 7*log(2)^2*zeta(3)/4 - 45*log(2)*zeta(4)/16

Crossrefs

Cf. A001008, A002805, A007408, A007409.

Cf. A002117, A002162, A013661, A013662, A013663.

Sequence in context: A010781 A019806 A199589 * A383432 A354977 A080172

Adjacent sequences: A398071 A398072 A398073 * A398077 A398078 A398079

Keyword

nonn,cons,new

Author

Amiram Eldar, Jul 19 2026

Status

approved