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A339097
Decimal expansion of Sum_{k>=1} zeta(4*k+1)-1.
2
0, 3, 9, 0, 6, 7, 0, 0, 7, 2, 3, 7, 9, 9, 5, 0, 8, 1, 0, 6, 0, 8, 0, 4, 7, 1, 3, 5, 9, 7, 8, 4, 3, 4, 2, 3, 2, 4, 0, 7, 8, 8, 8, 4, 6, 1, 4, 8, 2, 6, 7, 3, 8, 8, 9, 8, 0, 6, 2, 1, 5, 2, 0, 4, 9, 8, 1, 1, 3, 5, 7, 9, 2, 3, 1, 5, 2, 7, 3, 3, 7, 8, 3, 9, 7, 9, 1, 1, 1, 3, 6, 0, 6, 3, 9, 9, 7, 8, 9, 3, 3, 5, 8, 0, 1, 9
OFFSET
0,2
FORMULA
Equals Sum_{k>=2} (k^3 -3*k^2 + k - 2)/(k^5 - k).
Equals 3/8 - gamma/2 - Re(Psi(i))/2, where Psi is the digamma function, gamma is the Euler-Mascheroni constant (see A001620), and i=sqrt(-1).
Equals 3/8 - Re(H(I))/2, where H is the harmonic number function.
Equals 1/4 - A338858.
Equals Sum_{k>=2} 1/(k*(k^4 - 1)). - Vaclav Kotesovec, Dec 24 2020
EXAMPLE
0.0390670072379950810608...
MATHEMATICA
Join[{0}, RealDigits[N[Re[Sum[Zeta[4 n + 1] - 1, {n, 1, Infinity}]], 105]][[1]]]
PROG
(PARI) suminf(k=1, zeta(4*k+1)-1) \\ Michel Marcus, Dec 24 2020
CROSSREFS
Cf. A256919 (4*k), A339083 (4*k+2), A338858 (4k+3).
Sequence in context: A021260 A215633 A134878 * A154540 A324834 A019832
KEYWORD
nonn,cons
AUTHOR
Artur Jasinski, Dec 24 2020
STATUS
approved