A397003 Decimal expansion of Sum_{k>=1} H(k)/k^5, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

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

Offset

1,3

References

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

Jonathan Borwein and David Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century, A K Peters, 2004, p. 99.

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.

Christian Goldbach, Lettre LI, letter to Leonhard Euler, Dec. 24 1742, in P. H. Fuss, Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIème siècle, Vol. I, St.-Pétersbourg, 1843, pp. 176-177.

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

Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2nd ed., 2023, p. 252, eq. (4.6).

Michael I. Shamos, Shamos's Catalog of the Real Numbers, 2011, p. 105.

Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer International Publishing, 2019, section 3.10, p. 87, eq. (3.45), section 6.29, p. 419, eq. (6.132).

Eric Weisstein's World of Mathematics, Harmonic Number, eq. (21).

Index entries for sequences related to harmonic numbers.

Formula

Equals 7*zeta(6)/4 - zeta(3)^2/2.

Equals (19*Pi^6/5670 - Sum_{k>=1} H(k,2)/k^4)/2 = ((19/6)*A013664 - A238166)/2 (Goldbach, 1742).

Example

1.057879959255968877543563837730617010236312904675549...

Mathematica

RealDigits[7*Zeta[6]/4 - Zeta[3]^2/2, 10, 120][[1]]

Prog

(PARI) 7*zeta(6)/4 - zeta(3)^2/2

Crossrefs

Cf. A001008, A002805.

Cf. A002117, A013664, A238166.

Sum_{k>=1} H(k)/k^m: A076788 (m=1), A233090 (m=2), A233033 (m=3), A396737 (m=4), this constant (m=5), A397004 (m=6), A397005 (m=7), A397006 (m=8), A397007 (m=9), A397008 (m=10).

Sequence in context: A385857 A338595 A394469 * A254274 A068001 A213443

Adjacent sequences: A397000 A397001 A397002 * A397004 A397005 A397006

Keyword

nonn,cons

Author

Amiram Eldar, Jun 13 2026

Status

approved