A391304 Decimal expansion of Sum_{k>=1} Sum_{m>=1} (-1)^(m+k) * H(m+k)/(m+k), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

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

Offset

0,1

References

Hongwei Chen, Monthly Problem Gems, CRC Press, 2021, p. 103.

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals, Springer, 2013, section 3.103, p. 157.

Links

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

Ovidiu Furdui, Problem 11519, The American Mathematical Monthly, Vol. 117, No. 7 (2010), p. 649; A Harmonious Sum, Solutions to Problem 11519, by Wim Nuij and Richard Stong, ibid., Vol. 119, No. 5 (2012), pp. 431-433.

Ovidiu Furdui, Closed form evaluation of a multiple harmonic series (abstract only), Automat. Comput. Appl. Math., Vol. 20, No. 1 (2011), pp. 19-24.

Ovidiu Furdui and Tiberiu Trif, On the Summation of Certain Iterated Series, Journal of Integer Sequences, Vol. 14 (2011), Article 11.6.1.

Richard J. Mathar, Yet Another Table of Integrals, arXiv:1207.5845 [math.CA], 2012-2024, eq. (0.410), p. 42.

Formula

Equals Pi^2/12 - log(2)/2 - log(2)^2/2.

Example

0.235666936185039851194040259430591824706448407625998...

Mathematica

RealDigits[Pi^2/12 - Log[2]/2 - Log[2]^2/2, 10, 120][[1]]

Prog

(PARI) Pi^2/12 - log(2)/2 - log(2)^2/2

Crossrefs

Cf. A001008, A002805.

Cf. A002162, A072691, A253191.

Similar sums: A187832 (without H(m+k)), A256358 (with log(m+k) instead of H(m+k)).

Sequence in context: A316609 A307327 A000036 * A165081 A165089 A165083

Adjacent sequences: A391301 A391302 A391303 * A391305 A391306 A391307

Keyword

nonn,cons,easy

Author

Amiram Eldar, Feb 09 2026

Status

approved