A390210 Decimal expansion of Sum_{k>=1} (-1)^(k+1) * k * (H(k) - log(k) - gamma - 1/(2*k)), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number (negated).

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

Offset

0,2

References

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See Problem 3.43, pages 145, 165 and 195.

Links

Table of n, a(n) for n=0..105.

Vincent Nguyen, On some series involving harmonic and skew-harmonic numbers, Journal of Classical Analysis, Vol. 22, No. 1 (2023), pp. 25-43; arXiv preprint, arXiv:2304.11614 [math.CA], 2023-2024. See eq. (3.3), p. 38 (p. 14 in the preprint).

Formula

Equals 3*log(A) - (gamma + 1)/4 - 7*log(2)/12, where A is the Glaisher-Kinkelin constant (A074962).

Example

-0.05237634045066519133658777930952834417913910468952...

Mathematica

RealDigits[3 * Log[Glaisher] - (EulerGamma + 1)/4 - 7*Log[2]/12, 10, 120, -1][[1]]

Prog

(PARI) 1/4 - 3*zeta'(-1) - (Euler + 1)/4 - 7*log(2)/12

Crossrefs

Cf. A001008, A002805.

Cf. A001620 (gamma), A074962 (A), A225746 (log(A)).

Similar constants: A256921, A345208.

Sequence in context: A063572 A205294 A372806 * A131567 A200301 A114746

Adjacent sequences: A390207 A390208 A390209 * A390211 A390212 A390213

Keyword

nonn

Author

Amiram Eldar, Oct 29 2025

Status

approved