A386442 Decimal expansion of Sum_{k>=2} H(k) * (zeta(k) - 1), where H(k) is the k-th harmonic number.

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

Offset: 1

Author

Amiram Eldar, Jul 21 2025

Keywords

• nonn
• cons

Examples

			1.68053122204283676940338774041347909746938155456896...
		

Links

• Khristo N. Boyadzhiev, A special constant and series with zeta values and harmonic numbers, arXiv:1903.11141 [math.NT], 2019.
• Michael I. Shamos, Shamos's catalog of the real numbers, 2011. See p. 565.

Crossrefs

Cf. A001620, A001008, A002805, A131688.

Programs

Mathematica

RealDigits[NIntegrate[HarmonicNumber[x]/x, {x, 0, 1}, WorkingPrecision -> 120] + 1 - EulerGamma][[1]]

PARI

sumpos(k = 2, (k/(k-1))*log(k/(k-1)) - 1/k)

Formula

Equals 1 + Sum_{k>=2} (H(k) - 1) * (zeta(k) - 1).

Equals Sum_{k>=2} ((k/(k-1))*log(k/(k-1)) - 1/k) (Shamos, 2011).

Equals 1 + Sum_{k>=2} (-1)^k * zeta(k) / (k*(k-1)) (Shamos, 2011).

Equals M + 1 - gamma, where M = A131688 and gamma = A001620 (Boyadzhiev, 2019).