A397005 Decimal expansion of Sum_{k>=1} H(k)/k^7, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

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

Offset

1,4

References

Bruce C. Berndt, Ramanujan's Notebooks Part 1, Springer Verlag, 1985, pp. 252-253, eq. (9.5).

H. M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Insights, 2011, p. 228, problem 14.

Links

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

David Borwein and Jonathan M. Borwein, On an intriguing integral and some series related to zeta(4), Proc. Amer. Math. Soc., Vol. 123, No. 4 (1995), pp. 1191-1198. See p. 1196.

Leonhard Euler, Meditationes circa singulare serierum genus, Novi Commentarii academiae scientiarum Petropolitanae, Vol. 20 (1776), pp. 140-186.

C. Georghiou and A. N. Philippou, Harmonic sums and the Zeta function, Fibonacci Quarterly, Vol. 21, No. 1 (1983), pp. 29-36.

Ross C. McPhedran and David H. Bailey, New Results for Euler Sums, arXiv:2311.06294 [math.NT], 2023-2025.

Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer International Publishing, 2019, section 3.10, p. 87, eq. (3.45).

Eric Weisstein's World of Mathematics, Harmonic Number, eq. (24).

Index entries for sequences related to harmonic numbers.

Formula

Equals 9*zeta(8)/4 - zeta(3)*zeta(5).

Example

1.012727885297505444879561647888274487841557977977895...

Mathematica

RealDigits[9*Zeta[8]/4 - Zeta[3]*Zeta[5], 10, 120][[1]]

Prog

(PARI) 9*zeta(8)/4 - zeta(3)*zeta(5)

Crossrefs

Cf. A001008, A002805.

Cf. A002117, A013663, A013666.

Sum_{k>=1} H(k)/k^m: A076788 (m=1), A233090 (m=2), A233033 (m=3), A396737 (m=4), A397003 (m=5), A397004 (m=6), this constant (m=7), A397006 (m=8), A397007 (m=9), A397008 (m=10).

Sequence in context: A181284 A010697 A088666 * A170854 A215140 A278419

Adjacent sequences: A397002 A397003 A397004 * A397006 A397007 A397008

Keyword

nonn,cons

Author

Amiram Eldar, Jun 13 2026

Status

approved