A397630 Decimal expansion of Sum_{k>=1} H(k)^2*H(k,3)/k^2, 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 generalized harmonic number of order 3.

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

Offset

1,1

Links

Table of n, a(n) for n=1..105.

Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer International Publishing, 2019, section 4.43, pp. 302-303, eq. (4.59), section 5.43, p. 324, section 6.43, pp. 470-479.

Index entries for sequences related to harmonic numbers.

Formula

Equals 329*zeta(7)/16 - 9*zeta(2)*zeta(5)/2 - 6*zeta(3)*zeta(4).

Example

5.252547277139822199775481525627052538019850512446406...

Mathematica

RealDigits[329*Zeta[7]/16 - 9*Zeta[2]*Zeta[5]/2 - 6*Zeta[3]*Zeta[4], 10, 120][[1]]

Prog

(PARI) 329*zeta(7)/16 - 9*zeta(2)*zeta(5)/2 - 6*zeta(3)*zeta(4)

Crossrefs

Cf. A001008, A002805, A007408, A007409.

Cf. A002117, A013661, A013662, A013663, A013665.

Sequence in context: A382579 A059688 A072996 * A244892 A278066 A153107

Adjacent sequences: A397625 A397626 A397629 * A397631 A397632 A397633

Keyword

nonn,cons,new

Author

Amiram Eldar, Jul 03 2026

Status

approved