A210593 -id:A210593 - cp's OEIS frontend

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

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.

A349252 Decimal expansion of Sum_{k>=1} (-1)^k * log(k) / k^4.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Nov 12 2021

First derivative of the Dirichlet eta function at 4.

			0.0334788045785650663859568547887377997137597304057...

Eric Weisstein's World of Mathematics, Dirichlet Eta Function

Cf. A013662, A091812, A210593, A256358, A261506, A267315, A349220.

Flatten[{0, RealDigits[(Pi^4 Log[2] + 630 Zeta'[4])/720, 10, 120][[1]]}]

sumalt(k=1, (-1)^k * log(k) / k^4) \\ Michel Marcus, Nov 12 2021

Equals (Pi^4 * log(2) + 630 * zeta'(4)) / 720.

A373208 Decimal expansion of Product_{k>=1} f(2k)^2/(f(2k-1) * f(2*k+1)), where f(k) = k^(1/k^2).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 28 2024

			(2^(1/2^2)/1^1^2) * (2^(1/2^2)/3^(1/3^2)) * (4^(1/4^2)/3^(1/3^2)) * (4^(1/4^2)/5^(1/5^2)) * ...
1.22462314058511114559525704516215894720101844832032...

Dirk Huylebrouck, Generalizing Wallis' formula, The American Mathematical Monthly, Vol. 122, No. 4 (2015), pp. 371-372; alternative link; arXiv preprint, arXiv:1402.6577 [math.HO], 2014.
Eric Weisstein's World of Mathematics, Dirichlet Eta Function.
Wikipedia, Dirichlet eta function.

Cf. A001620, A013661, A073004, A074962, A210593, A373207.

RealDigits[(4 * Pi * Exp[EulerGamma] / Glaisher^12)^Zeta[2], 10, 120][[1]]

(4 * Pi * exp(Euler - 1 + 12*zeta'(-1)))^zeta(2)

Equals exp(2*eta'(2)) = exp(2*A210593), where eta is the Dirichlet eta function.

Equals (4*Pi*exp(gamma)/A^12)^zeta(2), where gamma is Euler's constant (A001620) and A is the Glaisher-Kinkelin constant (A074962).

A375506 Decimal expansion of the first derivative of the Dirichlet eta-function eta(s) at s=3/2.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Aug 18 2024

			0.12867475083035719009595292991030137571142185354249...

Cf. A091812 (at s=1), A210593 (at s=2), A349220 (at s=3), A078434 (zeta(3/2)), A375503 (zeta'(3/2)).

s :=3/2 ; 2^(1-s)*log(2)*Zeta(s)+(1-2^(1-s))*Zeta(1,s) ; evalf(%) ;

RealDigits[DirichletEta'[3/2], 10, 120][[1]] (* Amiram Eldar, Aug 19 2024 *)

Equals log(2)*zeta(3/2)/sqrt(2) +(1-1/sqrt(2))*zeta'(3/2) = Sum_{i>=1} (-1)^i*log(i)/i^(3/2).

A369885 Decimal expansion of Sum_{k>=1} log(k+1)/k^2.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Feb 04 2024

			1.80075505600528299149660601421484318144566378381841...

Bernard Candelpergher and Marc-Antoine Coppo, A new class of identities involving Cauchy numbers, harmonic numbers and zeta values, The Ramanujan Journal, Vol. 27 (2012), pp. 305-328; alternative link.
István Mező, Problem 11793, Problems and Solutions, The American Mathematical Monthly, Vol. 121, No. 7 (2014), p. 648; A Series with Zetas, Solution to Problem 11793 by FAU Problem Solving Group, ibid., Vol. 123, No. 6 (2016), p. 620.
Michael Ian Shamos, A catalog of the real numbers (2011), p. 616.
Roberto Tauraso, Problem 11793.
Wikipedia, Harmonic number: Harmonic numbers for real and complex values.

Cf. A001008, A001620, A002805, A006232, A006233, A073002, A210593.

evalf(sum((-1)^(k+1)*Zeta(k)/(k-2), k = 3 .. infinity) - Zeta(1, 2), 120)

RealDigits[NIntegrate[HarmonicNumber[x]/x^2, {x, 1, Infinity}, WorkingPrecision -> 120]][[1]]

PARI
```
sumpos(k = 1, log(k+1)/k^2)
```

sumalt(k = 3, (-1)^(k+1) * zeta(k)/(k-2)) - zeta'(2)

Equals Integral_{x>=1} H(x)/x^2 dx, where H(x) is the harmonic number for real variable x (Shamos, 2011).
Equals -zeta'(2) + Sum_{k>=3} (-1)^(k+1)*zeta(k)/(k-2) (Mező, 2014).
Equals Sum_{k>=1} lambda(k)*H(k)/(k^2*k!) + 1 + zeta(3) - gamma * zeta(2), where lambda(k) = abs(A006232(k)/A006233(k)) is the n-th non-alternating Cauchy number, H(k) = A001008(k)/A002805(k) is the k-th harmonic number, and gamma is Euler's constant (A001620) (Candelpergher and Coppo, 2012). - Amiram Eldar, Mar 18 2024

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

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A349252 Decimal expansion of Sum_{k>=1} (-1)^k * log(k) / k^4.

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A373208 Decimal expansion of Product_{k>=1} f(2*k)^2/(f(2*k-1) * f(2*k+1)), where f(k) = k^(1/k^2).

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A375506 Decimal expansion of the first derivative of the Dirichlet eta-function eta(s) at s=3/2.

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A369885 Decimal expansion of Sum_{k>=1} log(k+1)/k^2.

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A373208 Decimal expansion of Product_{k>=1} f(2k)^2/(f(2k-1) * f(2*k+1)), where f(k) = k^(1/k^2).