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

%I A339604 #50 Jan 15 2021 13:34:47
%S A339604 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,
%T A339604 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,
%U A339604 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
%N A339604 Decimal expansion of Sum_{k>=1} (zeta(3*k)-1).
%C A339604 For additional comments and generalization see attached text file.
%H A339604 Artur Jasinski, <a href="/A339604/a339604.txt">Sums of zeta(p*n+q)-1</a>
%H A339604 T. J. Stieltjes, <a href="https://projecteuclid.org/euclid.acta/1485881121">Table des valeurs des sommes Sk</a>, Acta Math., Volume 10 (1887), 299-302.
%F A339604 Equals Sum_{k>=2} 1/(k^3-1).
%F A339604 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).
%F A339604 Equals 1 + gamma/3 - (1/3)*A339135 + 2*log(2)/9 - sqrt(3)*Pi*tanh(sqrt(3)*Pi/2)/6.
%F A339604 Equals 7/6 - Pi*tanh(Pi*sqrt(3)/2)/(2*sqrt(3)) - A339605/2.
%F A339604 Equals 4/3 - Pi*tanh(Pi*sqrt(3)/2)/sqrt(3) + A339606.
%F A339604 Equals 1 - A339605 - A339606.
%e A339604 0.221689395109267...
%t A339604 RealDigits[Chop[N[Sum[Zeta[3 n] - 1, {n, 1, Infinity}], 105]]][[1]]
%o A339604 (PARI) suminf(k=1, zeta(3*k)-1) \\ _Michel Marcus_, Dec 09 2020
%Y A339604 Cf. A024006, A256919, A338815, A339135, A339529, A339530, A339605, A339606.
%K A339604 nonn,cons
%O A339604 0,1
%A A339604 _Artur Jasinski_, Dec 09 2020