A339604 Decimal expansion of Sum_{k>=1} (zeta(3*k)-1).

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

Offset

0,1

Comments

For additional comments and generalization see attached text file.

Links

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

Artur Jasinski, Sums of zeta(p*n+q)-1

T. J. Stieltjes, Table des valeurs des sommes Sk, Acta Math., Volume 10 (1887), 299-302.

Formula

Equals Sum_{k>=2} 1/(k^3-1).

Equals 1 + gamma/3 + (1/3)*Re(Psi(1/2 + i*sqrt(3)/2)) - sqrt(3)*Pi*tanh(sqrt(3)*Pi/2)/6, where Psi is the digamma function, gamma is the Euler-Mascheroni constant (see A001620), and i=sqrt(-1).

Equals 1 + gamma/3 - (1/3)*A339135 + 2*log(2)/9 - sqrt(3)*Pi*tanh(sqrt(3)*Pi/2)/6.

Equals 7/6 - Pi*tanh(Pi*sqrt(3)/2)/(2*sqrt(3)) - A339605/2.

Equals 4/3 - Pi*tanh(Pi*sqrt(3)/2)/sqrt(3) + A339606.

Equals 1 - A339605 - A339606.

Example

0.221689395109267...

Mathematica

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

Prog

(PARI) suminf(k=1, zeta(3*k)-1) \\ Michel Marcus, Dec 09 2020

Crossrefs

Cf. A024006, A256919, A338815, A339135, A339529, A339530, A339605, A339606.

Sequence in context: A118024 A184184 A074297 * A020824 A362708 A138678

Adjacent sequences: A339601 A339602 A339603 * A339605 A339606 A339607

Keyword

nonn,cons

Author

Artur Jasinski, Dec 09 2020

Status

approved