A339529 -id:A339529 - 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

A339530 Decimal expansion of Sum_{k>=1} (zeta(8*k)-1).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Dec 08 2020, following a suggestion of Artur Jasinski

			0.00409269829928628730747620468964025986524982473540016981249105600555721...

Cf. A024006, A256919, A339529.

Join[{0, 0}, RealDigits[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])), 10, 100][[1]]]

Equals Sum_{k>=2} 1/(k^8 - 1).
Equals 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))).
Equals (1/2)*Sum_{k>=2} 1/(k^4-1) - (1/2)*Sum_{k>=2} 1/(k^4+1) = (A256919-A256920)/2. - R. J. Mathar, Jan 22 2021

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

A024004 a(n) = 1 - n^6.

Original entry on oeis.org

1, 0, -63, -728, -4095, -15624, -46655, -117648, -262143, -531440, -999999, -1771560, -2985983, -4826808, -7529535, -11390624, -16777215, -24137568, -34012223, -47045880, -63999999, -85766120, -113379903, -148035888, -191102975, -244140624, -308915775, -387420488, -481890303

Offset: 0

N. J. A. Sloane, corrected Mar 01 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A001014.

a(n) = -A123866(n) for n > 0.

a024004 = (1 -) . (^ 6)  -- Reinhard Zumkeller, Mar 11 2014

[1-n^6: n in [0..50]]; // Vincenzo Librandi, Apr 29 2011

Table[1-n^6,{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)

A024004(n):=1-n^6 $ makelist(A024004(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

for(n=0,50, print1(1-n^6, ", ")) \\ G. C. Greubel, May 11 2017

From G. C. Greubel, May 11 2017: (Start)
G.f.: (1 - 7*x - 42*x^2 - 322*x^3 - 287*x^4 - 63*x^5)/(1 - x)^7.
E.g.f.: (1 - x - 31*x^2 - 90*x^3 - 65*x^4 - 15*x^5 - x^6)*exp(x). (End)
Sum_{k>=2} -1/a(k) = 11/12 - Pi*tanh(sqrt(3)*Pi/2)/(2*sqrt(3)) = A339529. - Vaclav Kotesovec, Dec 08 2020

A339520 Odd composite integers m such that A086902(2m-J(m,53)) == 7J(m,53) (mod m), where J(m,53) is the Jacobi symbol.

Original entry on oeis.org

25, 35, 51, 65, 75, 91, 105, 175, 203, 325, 391, 455, 575, 645, 861, 1247, 1275, 1295, 1633, 1763, 1775, 1785, 1875, 1921, 2275, 2407, 2415, 2599, 2625, 2651, 3045, 3367, 4199, 4579, 4623, 5629, 5835, 5887, 6441, 6699, 9959, 10465, 10815, 10825, 10877, 11865, 12025

Offset: 1

Ovidiu Bagdasar, Dec 07 2020

nonn

The generalized Pell-Lucas sequences of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy V(k*p-J(p,D)) == V(k-1)*J(p,D) (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4.
The composite integers m with the property V(k*m-J(m,D)) == V(k-1)*J(m,D) (mod m) are called generalized Pell-Lucas pseudoprimes of level k- and parameter a.
Here b=-1, a=7, D=53 and k=2, while V(m) recovers A086902(m).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 24(1), 9-15 (2018).

Cf. A086902, A071904, A339128 (a=7, b=-1, k=1).

Cf. A339517 (a=1, b=-1), A339518 (a=3, b=-1), A339529 (a=5, b=-1).

Select[Range[3, 20000, 2], CoprimeQ[#, 53] && CompositeQ[#] && Divisible[LucasL[2*# - JacobiSymbol[#, 53], 7] - 7*JacobiSymbol[#, 53], #] &]

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

cp's OEIS Frontend

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

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A339530 Decimal expansion of Sum_{k>=1} (zeta(8*k)-1).

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

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A024004 a(n) = 1 - n^6.

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A339520 Odd composite integers m such that A086902(2*m-J(m,53)) == 7*J(m,53) (mod m), where J(m,53) is the Jacobi symbol.

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

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A339520 Odd composite integers m such that A086902(2m-J(m,53)) == 7J(m,53) (mod m), where J(m,53) is the Jacobi symbol.