A397030 - OEIS

%I #5 Jun 14 2026 11:36:15

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

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

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

%N Decimal expansion of Sum_{k>=1} H(k) / (2*k+1)^4, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

%H Ross C. McPhedran and David H. Bailey, <a href="https://arxiv.org/abs/2311.06294">New Results for Euler Sums</a>, arXiv:2311.06294 [math.NT], 2023-2025. See p. 19, eq. (106).

%H Ali Shadhar Olaikhan, <a href="https://www.researchgate.net/publication/373488370">An Introduction to the Harmonic Series and Logarithmic Integrals</a>, 2nd ed., 2023, section 4.1.14, pp. 273-275.

%H Cornel Ioan Vălean, <a href="https://doi.org/10.1007/978-3-031-21262-8">More (Almost) Impossible Integrals, Sums, and Series</a>, Springer Cham, 2023. See section 4.19, pp. 419-420, eq.(4.101), section 5.19, pp. 460-461, and section 6.19, pp. 595-601.

%H <a href="/index/Ha#harmonic">Index entries for sequences related to harmonic numbers</a>.

%F Equals 31*zeta(5)/8 - 21*zeta(2)*zeta(3)/16 - 15*log(2)*zeta(4)/8.

%e 0.016240657850235703095128372905552235813003886805619...

%t RealDigits[31*Zeta[5]/8 - 21*Zeta[2]*Zeta[3]/16 - 15*Log[2]*Zeta[4]/8, 10, 120, -1][[1]]

%o (PARI) 31*zeta(5)/8 - 21*zeta(2)*zeta(3)/16 - 15*log(2)*zeta(4)/8

%Y Cf. A001008, A002805.

%Y Cf. A002117, A002162, A013662.

%Y Sum_{k>=1} H(k) / (2*k+1)^m: A397028 (m=2), A397029 (m=3), this constant (m=4), A397031 (m=5), A397032 (m=6).

%K nonn,cons

%O 0,3

%A _Amiram Eldar_, Jun 14 2026