A197070 -id:A197070 - cp's OEIS frontend

A334582 Denominator of Sum_{k=1..n} (-1)^(k+1)/k^3.

1, 8, 216, 1728, 216000, 216000, 74088000, 592704000, 16003008000, 16003008000, 21300003648000, 21300003648000, 46796108014656000, 6685158287808000, 6685158287808000, 53481266302464000, 262753461344005632000, 262753461344005632000

Offset: 1

Petros Hadjicostas, May 06 2020

For n = 1 to n = 13, a(n) = A195506(n), but a(14) = 6685158287808000 <> 46796108014656000 = A195506(14).

Lim_{n -> infinity} A136675(n)/a(n) = A197070.

			The first few fractions are 1, 7/8, 197/216, 1549/1728, 195353/216000, 194353/216000, 66879079/74088000, 533875007/592704000, ... = A136675/A334582.

Wikipedia, Dirichlet eta function.

Cf. A136675 (numerators), A195506, A197070.

b := proc(n) local k: add((-1)^(k + 1)/k^3, k = 1 .. n): end proc:
seq(denom(b(n)), n=1..30);

Denominator @ Accumulate[Table[(-1)^(k + 1)/k^3, {k, 1, 18}]] (* Amiram Eldar, May 08 2020 *)

a(n) = denominator(sum(k=1, n, (-1)^(k+1)/k^3)); \\ Michel Marcus, May 07 2020

Offset changed to 1 by Georg Fischer, Jul 13 2023

A347059 Decimal expansion of the Dirichlet eta function at 9.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Aug 14 2021

			0.998094297541605330767783031852597950...

L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (306).

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A072691, A197070, A267315, A267316, A275703, A275710, A347150, A346927.

First[RealDigits[N[DirichletEta[9],87]]] (* Stefano Spezia, Aug 15 2021 *)

-polylog(9, -1) \\ Michel Marcus, Aug 15 2021

Equals (255/256) * zeta(9).
Equals Sum_{k>=1} (-1)^(k+1) / k^9.
Equals eta(9).

A347150 Decimal expansion of the Dirichlet eta function at 8.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Aug 19 2021

			0.9962330018526478992272892600828036178741251594728980...

L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (306).

Michael I. Shamos, Shamos's catalog of the real numbers (2011).
Index entries for transcendental numbers

Cf. A072691, A197070, A267315, A267316, A275703, A275710, A346927, A347059.

RealDigits[DirichletEta[8], 10, 100][[1]] (* Amiram Eldar, Aug 20 2021 *)

-polylog(8, -1) \\ Michel Marcus, Aug 20 2021

Equals (127/128) * zeta(8).
Equals 127 * Pi^8 / 1209600.
Equals Sum_{k>=1} (-1)^(k+1) / k^8.
Equals eta(8).

A349220 Decimal expansion of Sum_{k>=1} (-1)^k * log(k) / k^3.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Nov 11 2021

First derivative of the Dirichlet eta function at 3.

			0.0597059061601953583634292662879256783169268731565...

Eric Weisstein's World of Mathematics, Dirichlet Eta Function

Cf. A002117, A091812, A197070, A210593, A244115, A256358.

Flatten[{0, RealDigits[(Log[2] Zeta[3] + 3 Zeta'[3])/4, 10, 120][[1]]}]

sumalt(k=1, (-1)^k * log(k) / k^3) \\ Michel Marcus, Nov 11 2021

Equals (log(2) * zeta(3) + 3 * zeta'(3)) / 4.

A097449 If n is a cube, replace it with the cube root of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 2, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 3, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 4, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 0

Cino Hilliard, Aug 23 2004

			The 9th integer is 8 so a(9) = 8^(1/3) = 2.

Cf. A016627, A097448, A197070.

rcr[n_]:=Module[{crn=Power[n, (3)^-1]},If[IntegerQ[crn],crn,n]]; Array[ rcr,80,0] (* Harvey P. Dale, Jan 28 2012 *)

iscube(n) = { local(r); r = n^(1/3); if(floor(r+.5)^3== n,1,0) }
replcube(n) = { for(x=0,n, if(iscube(x),y=x^(1/3),y=x); print1(floor(y)",")) }

a(n)=ispower(n,3,&n);n \\ Charles R Greathouse IV, Oct 27 2011

Sum_{n>=1} (-1)^(n+1)/n = 2*log(2) - 3*zeta(3)/4 = A016627 - A197070. - Amiram Eldar, Jul 07 2024

Corrected by T. D. Noe, Oct 25 2006

A346927 Decimal expansion of the Dirichlet eta function at 10.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Aug 07 2021

			0.999039507598271565639221845699341831425929649666890...

L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (306).

Michael I. Shamos, Shamos's catalog of the real numbers (2011).
Index entries for transcendental numbers

Cf. A072691, A197070, A267315, A267316, A275703, A275710, A347150, A347059.

RealDigits[DirichletEta[10], 10, 100][[1]] (* Amiram Eldar, Aug 08 2021 *)

-polylog(10, -1) \\ Michel Marcus, Aug 08 2021

Equals 73 * Pi^10 / (2^9 * 3^5 * 5 * 11).
Equals (511/512) * zeta(10).
Equals Sum_{k>=1} (-1)^(k+1) / k^10.
Equals eta(10).

A275689 Decimal expansion of 3zeta(3)/(4log(2)).

Original entry on oeis.org

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

Offset: 1

Terry D. Grant, Aug 05 2016

As it appears that the Sum {n>=1} (-1)^(n+1)/n^2 / Sum {n>=1} ((-1)^(n+1))/n^1 is the inverse of Levy's constant, or more traditionally the log of Levy's constant (A100199), this sequence which is equal to Sum {n>=1} (-1)^(n+1)/n^3 / Sum {n>=1} ((-1)^(n+1))/n^1 may be the inverse of the log of another constant with similar properties.

			1.300651149791018703323...

Cf. A197070, A002162, A100199.

RealDigits[3*Zeta[3]/(4*Log[2]), 10, 120][[1]] (* Amiram Eldar, May 27 2023 *)

3*zeta(3)/log(16) \\ Charles R Greathouse IV, Aug 05 2016

sumalt(n=1,(-1)^n/n^3)/sumalt(n=1,(-1)^n/n) \\ Charles R Greathouse IV, Aug 05 2016

3*zeta(3)/4*log(2) = A197070 / A002162 = Sum {n>=1} (-1)^(n+1)/n^3 / Sum {n>=1} ((-1)^(n+1))/n^1

A246967 Decimal expansion of the real positive solution to eta(x) = x.

Original entry on oeis.org

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

Offset: 0

Michal Paulovic, Sep 08 2014

Fixed point of Dirichlet eta function.

			0.6293340940...

Eric Weisstein's World of Mathematics, Dirichlet Eta Function.
Wikipedia, Dirichlet Eta Function.

Cf. A197070, A267315, A267316.

RealDigits[x /. FindRoot[DirichletEta[x] - x, {x, 0}, WorkingPrecision -> 120], 10, 100] [[1]] (* Amiram Eldar, May 24 2021 *)

solve(n=0,2,(1-2^(1-n))*zeta(n)-n) \\ Edward Jiang, Sep 08 2014

A269444 Continued fraction expansion of the Dirichlet eta function at 3.

Original entry on oeis.org

0, 1, 9, 6, 2, 1, 1, 1, 1, 1, 1, 6, 1, 4, 1, 7, 2, 1, 1, 1, 2, 91, 32, 1, 1, 6, 23, 1, 1, 1, 1, 2, 9, 1, 2, 1, 1, 5, 1, 1, 37, 12, 1, 12, 3, 2, 87, 1, 4, 2, 2, 2, 320, 1, 7, 1, 2, 6, 3, 1, 6, 4, 1, 4, 2, 1, 69, 1, 4, 3, 3, 1, 14, 3, 1, 3, 1, 10, 2, 694, 2, 4, 21, 1, 1, 1, 3, 3, 10, 2, 1, 2, 2, 1, 3, 5, 1, 3, 9, 1

Offset: 0

Ilya Gutkovskiy, Feb 26 2016

Continued fraction expansion of Sum_{k>=1} (-1)^(k - 1)/k^3 = (3*zeta(3))/4 = 0.901542677369695714...

			1/1^3 - 1/2^3 + 1/3^3 - 1/4^3 + 1/5^3 - 1/6^3 +... = 1/(1 + 1/(9 + 1/(6 + 1/(2 + 1/(1 + 1/(1 + 1/...)))))).

OEIS Wiki, Euler's alternating zeta function
Eric Weisstein's World of Mathematics, Dirichlet Eta Function
Wikipedia, Dirichlet Eta Function
Index entries for continued fractions for constants

Cf. A013631, A197070.

Mathematica
```
ContinuedFraction[(3 Zeta[3])/4, 100]
```

A275688 Decimal expansion of 9zeta(3)/(Pi^2log(2)).

Original entry on oeis.org

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

Offset: 1

Terry D. Grant, Aug 05 2016

			9*Zeta(3)/(Pi^2*log(2)) = 1.5814021680311229....

Cf. A002162, A100199, A197070.

RealDigits[9 Zeta[3]/(Pi^2 Log[2]),10,120][[1]] (* Harvey P. Dale, Jun 06 2022 *)

9*zeta(3)/Pi^2/log(2) \\ Charles R Greathouse IV, Aug 05 2016

9*zeta(3)/(Pi^2*log(2)) = ((3*zeta(3))/4)/((Pi^2)/(12*log(2))) =

A197070/A100199 = [Sum {n>=1} (-1)^(n+1)/n^3] / [( Sum_{n>=1} (-1)^(n+1)/n^2 ) / ( Sum_{n>=1} (-1)^(n+1)/n )].

Corrected and extended by Rick L. Shepherd, Nov 23 2016

Previous Mathematica program replaced by Harvey P. Dale, Jun 06 2022

cp's OEIS Frontend

A334582 Denominator of Sum_{k=1..n} (-1)^(k+1)/k^3.

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A347059 Decimal expansion of the Dirichlet eta function at 9.

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A347150 Decimal expansion of the Dirichlet eta function at 8.

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A349220 Decimal expansion of Sum_{k>=1} (-1)^k * log(k) / k^3.

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A097449 If n is a cube, replace it with the cube root of n.

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A346927 Decimal expansion of the Dirichlet eta function at 10.

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A275689 Decimal expansion of 3*zeta(3)/(4*log(2)).

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A275689 Decimal expansion of 3zeta(3)/(4log(2)).

A275688 Decimal expansion of 9zeta(3)/(Pi^2log(2)).