A339605 Decimal expansion of Sum_{k>=1} (zeta(3*k+1) - 1).

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

Offset

0,2

Links

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

Formula

Equals Sum_{k>=2} (k^5 - 3*k^4 + k^3 - k^2 + k - 1)/(k*(k^6 - 1)).

Equals 1/3 - 2*gamma/3 - (2/3)*Re(Psi(1/2 + i*sqrt(3)/2)), where Psi is the digamma function, gamma is the Euler-Mascheroni constant (see A001620), and i=sqrt(-1).

Equals 1/3 - 2*gamma/3 + 2*A339135/3 - 4*log(2)/9.

Equals 1 - A339604-A339606.

Equals Sum_{k>=2} 1/(k*(k^3 - 1)). - Vaclav Kotesovec, Dec 24 2020

Example

0.09180726255210907564327636663...

Mathematica

Join[{0}, RealDigits[Chop[N[Sum[Zeta[3 n + 1] - 1, {n, 1, Infinity}], 105]]][[1]]]

Prog

(PARI) suminf(k=1, zeta(3*k+1)-1) \\ Michel Marcus, Dec 09 2020
(PARI) sumnumrat(1/(x^4 - x), 2) \\ Charles R Greathouse IV, Jul 06 2026

Crossrefs

Cf. A256919, A338815, A339135, A339529, A339530, A339604, A339606, A001620.

Sequence in context: A084002 A376968 A385446 * A371856 A182494 A065444

Adjacent sequences: A339602 A339603 A339604 * A339606 A339607 A339608

Keyword

nonn,cons,changed

Author

Artur Jasinski, Dec 09 2020

Extensions

a(105) corrected by Georg Fischer, Nov 09 2025

Status

approved