A339530 -id:A339530 - cp's OEIS frontend

A256919 Decimal expansion of Sum_{k>=1} (zeta(4*k) - 1).

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

Offset: 0

Jean-François Alcover, Apr 13 2015

			0.0866629762657094129329746026249997547771718667980916672...
= -3 + Pi^4/90 + Pi^8/9450 + 691*Pi^12/638512875 + ...

H. M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Insights (2011) p. 265.

V. S. Adamchik, H. M. Srivastava, Some series of Zeta and related functions, Analysis (Munich) 18 (2) (1998) 131-144, eq. (2.25).
Eric Weisstein's MathWorld, Riemann Zeta Function
Wikipedia, Riemann Zeta Function

Cf. A013662, A013666, A013670, A175316, A256920, A339529, A339530, A339604 (3k-1).

Join[{0}, RealDigits[7/8 - (Pi/4)*Coth[Pi], 10, 104] // First]

Equals 7/8 - (Pi/4)*coth(Pi).
Equals Sum_{n>=2} 1/(n^4 - 1). - Vaclav Kotesovec, Dec 08 2020
Equals (1/2)* Sum_{n>=2} 1/(n^2-1) - (1/2)* Sum_{n>=2} 1/(n^2+1) = (3/4 - A100554)/2. - R. J. Mathar, Jan 22 2021

A339529 Decimal expansion of Sum_{k>=1} (zeta(6*k)-1).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Dec 08 2020

			0.017593026385321576213756604497955200206504349480747094973246930164332437...

Cf. A024004, A256919, A339530.

Join[{0}, RealDigits[11/12 - Pi*Tanh[Sqrt[3]*Pi/2]/(2*Sqrt[3]), 10, 100][[1]]]

Equals Sum_{k>=2} 1/(k^6 - 1).

Equals 11/12 - Pi*tanh(sqrt(3)*Pi/2)/(2*sqrt(3)).

A024006 a(n) = 1 - n^8.

Original entry on oeis.org

1, 0, -255, -6560, -65535, -390624, -1679615, -5764800, -16777215, -43046720, -99999999, -214358880, -429981695, -815730720, -1475789055, -2562890624, -4294967295, -6975757440, -11019960575, -16983563040, -25599999999, -37822859360, -54875873535

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..420
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A024004.

[1-n^8: n in [0..50]]; // Vincenzo Librandi, Apr 29 2011

Table[1-n^8,{n,0,40}] (* and *) LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {1, 0, -255, -6560, -65535, -390624, -1679615, -5764800, -16777215}, 40] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)

a(n)=1-n^8 \\ Charles R Greathouse IV, Oct 07 2015

Sum_{n>=2} -1/a(n) = 15/16 - Pi*(coth(Pi)/8) + Pi * (sin(sqrt(2)*Pi) + sinh(sqrt(2)*Pi)) / (4*sqrt(2) * (cos(sqrt(2)*Pi) - cosh(sqrt(2)*Pi))) = A339530 = 0.0040926982992862873... . - Vaclav Kotesovec, Feb 14 2015

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

Original entry on oeis.org

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

Artur Jasinski, Dec 09 2020

For additional comments and generalization see attached text file.

			0.221689395109267...

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.

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

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

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

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.

A339606 Decimal expansion of Sum_{k>=0} (zeta(3*k+2)-1).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Dec 09 2020

			0.6865033423386238859646...

Cf. A024006, A256919, A338815, A339135, A339529, A339530, A339604, A339605.

evalf(Re(sum(1/(k^3+1), k=1..infinity)), 120);  # Alois P. Heinz, Dec 12 2020

RealDigits[Chop[N[Sum[Zeta[3 n + 2] - 1, {n, 0, Infinity}], 105]]][[1]]

suminf(k=0, zeta(3*k+2)-1) \\ Michel Marcus, Dec 09 2020

Equals Sum_{k>=1} 1/(k^3 + 1).
Equals -1/3 + gamma/3 + (1/3)*Re(Psi(1/2 + i*sqrt(3)/2)) + sqrt(3)*Pi*tanh(sqrt(3)*Pi/2)/6, where Psi is digamma function, gamma is Euler-Mascheroni constant (see A001620), and i=sqrt(-1).
Equals -1/3 + gamma/3 - (1/3)*A339135 + 2*log(2)/9 + sqrt(3)*Pi*tanh(sqrt(3)*Pi/2)/6.
Equals 1 - A339605 - A339604.
Equals 1/2 + Sum_{k>=1} (-1)^(k+1) * (zeta(3*k)-1). - Amiram Eldar, Jan 07 2024

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

Original entry on oeis.org

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, 8

Offset: 0

Artur Jasinski, Dec 09 2020

			0.09180726255210907564327636663...

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

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

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

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

A338858 Decimal expansion of Sum_{k>=0} (zeta(4*k+3)-1).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Dec 24 2020

For additional comments and generalization see A339604.

			0.2109329927620049189391952864...

Cf. A256919 (4*k), A339097 (4*k+1), A339083 (4k+2).

Cf. A024006, A338815, A339135, A339529, A339530, A339604, A339605, A339606.

RealDigits[N[Re[Sum[Zeta[4 n + 3] - 1, {n, 0, Infinity}]], 105]][[1]]

suminf(k=0, zeta(4*k+3)-1) \\ Michel Marcus, Dec 24 2020

Equals Sum_{k>=2} k/(k^4-1).
Equals -1/8 + 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 -1/8 + Re(H(I))/2, where H is the harmonic number function.

A339083 Decimal expansion of Sum_{k>=0} (zeta(4*k+2)-1).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Dec 24 2020

Sum_{k>=1} zeta(4*k)-1 see A256919.
Sum_{k>=1} zeta(4*k+1)-1 see A339097.
Sum_{k>=0} zeta(4*k+3)-1 see A338858.
For additional comments and generalization see A339604.

			0.663337023734290587067025397375...

Cf. A024006, A256919, A338815, A338858, A339097, A339135, A339529, A339530, A339604, A339605, A339605.

RealDigits[N[Sum[Zeta[4 n + 2] - 1, {n, 0, Infinity}], 105]][[1]]

suminf(k=0, zeta(4*k+2)-1) \\ Michel Marcus, Dec 24 2020

Equals Sum_{k>=2} k^2/(k^4-1).

Equals -1/8 + Pi*coth(Pi)/4 = -1/8 + A338815 = 3/4 - A256919.

a(104) corrected and more terms from Georg Fischer, Jun 06 2024

A339097 Decimal expansion of Sum_{k>=1} zeta(4*k+1)-1.

Original entry on oeis.org

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

Artur Jasinski, Dec 24 2020

			0.0390670072379950810608...

Cf. A256919 (4*k), A339083 (4*k+2), A338858 (4k+3).

Cf. also A338815, A339135, A339529, A339530, A339604, A339605, A339606.

Join[{0},RealDigits[N[Re[Sum[Zeta[4 n + 1] - 1, {n, 1, Infinity}]], 105]][[1]]]

suminf(k=1, zeta(4*k+1)-1) \\ Michel Marcus, Dec 24 2020

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

A339801 Decimal expansion of the real part of harmonic number H(1/2 + i*sqrt(3)/2), where i=sqrt(-1).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Dec 17 2020

For imaginary part see A339802.

For real b, Im(Psi(1/2 + b*i)) = Pi*tanh(Pi*b)/2, but no such closed formula is known for the real part (see Wikipedia link). - Vaclav Kotesovec, Dec 19 2020

			0.862289106171836386535085450...

Wikipedia, Digamma function

Cf. A256919, A338815, A339135, A339529, A339530, A339604, A339605, A339606, A339802.

RealDigits[N[Re[HarmonicNumber[1/2 + I Sqrt[3]/2]], 105]][[1]]

Equals 1/2 + gamma + Re(Psi(1/2 + i*sqrt(3)/2)), where gamma is the Euler-Mascheroni constant (see A001620) and Psi is the digamma function.
Equals -1/2 + 3*A339604 + 3*A339606.
Equals Re((1 + i*sqrt(3))*Sum_{k>=0} 1/((1 + k)*(3 + i*sqrt(3) + 2*k))).

cp's OEIS Frontend

A256919 Decimal expansion of Sum_{k>=1} (zeta(4*k) - 1).

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A339529 Decimal expansion of Sum_{k>=1} (zeta(6*k)-1).

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A024006 a(n) = 1 - n^8.

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Magma

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

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A339606 Decimal expansion of Sum_{k>=0} (zeta(3*k+2)-1).

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Maple

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

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A338858 Decimal expansion of Sum_{k>=0} (zeta(4*k+3)-1).

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A339083 Decimal expansion of Sum_{k>=0} (zeta(4*k+2)-1).

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