A131688 -id:A131688 - cp's OEIS frontend

A379425 Decimal expansion of Ni_2 = gamma/3 - log(2Pi)/2 - 2zeta'(-1) + 2/3, where gamma = A001620.

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

Offset: 0

Artur Jasinski, Dec 22 2024

			0.270975642496740070183013614107411122680728399012594687451148817...

Marc-Antoine Coppo, A note on some alternating series involving zeta and multiple zeta values, Journal of Mathematical Analysis and Applications Volume 475, Issue 2, 15 July 2019, Pages 1831-1841.

Cf. A000796, A001620, A074962, A131688 (Ni_-1), A321943 (Ni_1), A379751 (Ni_3).

RealDigits[EulerGamma/3 - Log[2 Pi]/2 + 2/3 - 2 Zeta'[-1], 10, 105][[1]]

Equals Sum_{s>=2} (-1)^(s)*zeta(s)/(s+2).

Equals A001620/3 - log(2*A000796)/2 + 2*log(A074962) + 1/2.

A379751 Decimal expansion of Ni_3 = gamma/4 - log(2Pi)/2 - 3zeta'(-1) + 3*zeta'(-2) + 7/12, where gamma = A001620.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Jan 01 2025

Marc-Antoine Coppo, A note on some alternating series involving zeta and multiple zeta values, Journal of Mathematical Analysis and Applications Volume 475, Issue 2, 15 July 2019, Pages 1831-1841.

Cf. A000796, A001620, A074962, A131688 (Ni_-1), A321943 (Ni_1), A379425 (Ni_2).

RealDigits[EulerGamma/4 - Log[2 Pi]/2 - 3 Zeta'[-1] + 3 Zeta'[-2] + 7/12, 10, 105][[1]]

Equals Sum_{s>=2} (-1)^(s)*zeta(s)/(s+3).

A272286 Decimal expansion of Product_{k >= 1} (k(k+1))^(-1/(k(k+1))), a constant related to the alternating Lüroth representations of real numbers.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 24 2016

			0.1292150184060998413415719000742197771573364620386787448773...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8.1 Alternative representations [of real numbers], p. 62.

Sofia Kalpazidou, Khintchine's constant for Lüroth representation, Journal of Number Theory, Volume 29, Issue 2, June 1988, Pages 196-205.

Cf. A131688, A244109, A245254.

digits = 105; Exp[-NSum[((1 + (-1)^(n + 1))*Zeta[n + 1] - 1)/n, {n, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 2 digits, NSumTerms -> 200]] // RealDigits[#, 10, digits]& // First

Exp(-Sum_{n >= 1} (((1 + (-1)^(n+1))*Zeta(n+1) - 1)/n)). - After Vaclav Kotesovec's formula for A244109.

Offset corrected by Andrey Zabolotskiy, Dec 12 2023

A337128 Decimal expansion of Product_{k>=1} (1+1/k)^(1/k).

Original entry on oeis.org

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

Offset: 1

Michel Marcus, Sep 14 2020

			3.517487255902369649399793699323864170685620786764938...

Mathoverflow, Infinite product experimental mathematics question, Apr 21 2010.
Péter Pál Pach and Richárd Palincza, The counting version of a problem of Erdős, arXiv:2009.05305 [math.CO], 2020. Mentions this constant.

Cf. A131688.

First[RealDigits[Exp[NSum[Log[s]/(s(s-1)), {s, 2, Infinity}, NSumTerms -> 1000, Method -> {"NIntegrate", "MaxRecursion" -> 100}, WorkingPrecision -> 100]]]] (* Stefano Spezia, Jun 07 2024 *)

Equals exp(A131688). - Amiram Eldar, Sep 14 2020

More terms from Amiram Eldar, Sep 14 2020

A373862 Decimal expansion of Sum_{k >= 1} log(k)/(k*sqrt(k+1)).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jun 19 2024

			3.77100949140092...

Math StackExchange, How do I test for convergence of log(n)/n/sqrt(n+1), (2019)

Cf. A131688.

Digits := 120 ;
x := 0.0 ;
for l from 0 to 600 do
    x := x+(-1)^(l+1)*doublefactorial(2*l-1)/doublefactorial(2*l)*Zeta(1,3/2+l) ;
    x := evalf(x) ;
    print(x) ;
end do: # R. J. Mathar, Jun 27 2024

default(realprecision, 200); sumalt(k=0, (-1)^(k+1) * (2*k)! * zeta'(k+3/2) / (k!^2 * 4^k)) \\ Vaclav Kotesovec, Jun 27 2024

Equals sum_{l>=0} (-1)^(l+1) (2l-1)!! *Zeta'(3/2+l) /(2l)!!.

More terms from Vaclav Kotesovec, Jun 27 2024

A373863 Decimal expansion of Sum_{k>=1} log(k)/(k^2-k+1).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jun 19 2024

			1.0981067917544222069517...

Cf. A085609, A131688.

g := (1-e+e^2)^(-1) ;
x :=0.0 ;
for i from 0 to 350 do
    coeftayl(g,e=0,i) ;
    x-%*Zeta(1,2+i) ;
    x := evalf(%) ;
    print(%) ;
end do:

default(realprecision, 200); sumpos(k=1, log(k+1)/(k^2+k+1)) \\ Vaclav Kotesovec, Jun 28 2024

Equals Sum_{k>=1} log(k+1)/(k^2+k+1).

A386442 Decimal expansion of Sum_{k>=2} H(k) * (zeta(k) - 1), where H(k) is the k-th harmonic number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jul 21 2025

			1.68053122204283676940338774041347909746938155456896...

Khristo N. Boyadzhiev, A special constant and series with zeta values and harmonic numbers, arXiv:1903.11141 [math.NT], 2019.
Michael I. Shamos, Shamos's catalog of the real numbers, 2011. See p. 565.

Cf. A001620, A001008, A002805, A131688.

RealDigits[NIntegrate[HarmonicNumber[x]/x, {x, 0, 1}, WorkingPrecision -> 120] + 1 - EulerGamma][[1]]

sumpos(k = 2, (k/(k-1))*log(k/(k-1)) - 1/k)

Equals 1 + Sum_{k>=2} (H(k) - 1) * (zeta(k) - 1).
Equals Sum_{k>=2} ((k/(k-1))*log(k/(k-1)) - 1/k) (Shamos, 2011).
Equals 1 + Sum_{k>=2} (-1)^k * zeta(k) / (k*(k-1)) (Shamos, 2011).
Equals M + 1 - gamma, where M = A131688 and gamma = A001620 (Boyadzhiev, 2019).

cp's OEIS Frontend

A379425 Decimal expansion of Ni_2 = gamma/3 - log(2*Pi)/2 - 2*zeta'(-1) + 2/3, where gamma = A001620.

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A379751 Decimal expansion of Ni_3 = gamma/4 - log(2*Pi)/2 - 3*zeta'(-1) + 3*zeta'(-2) + 7/12, where gamma = A001620.

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A272286 Decimal expansion of Product_{k >= 1} (k*(k+1))^(-1/(k*(k+1))), a constant related to the alternating Lüroth representations of real numbers.

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A337128 Decimal expansion of Product_{k>=1} (1+1/k)^(1/k).

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A373862 Decimal expansion of Sum_{k >= 1} log(k)/(k*sqrt(k+1)).

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A373863 Decimal expansion of Sum_{k>=1} log(k)/(k^2-k+1).

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A386442 Decimal expansion of Sum_{k>=2} H(k) * (zeta(k) - 1), where H(k) is the k-th harmonic number.

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A379425 Decimal expansion of Ni_2 = gamma/3 - log(2Pi)/2 - 2zeta'(-1) + 2/3, where gamma = A001620.

A379751 Decimal expansion of Ni_3 = gamma/4 - log(2Pi)/2 - 3zeta'(-1) + 3*zeta'(-2) + 7/12, where gamma = A001620.

A272286 Decimal expansion of Product_{k >= 1} (k(k+1))^(-1/(k(k+1))), a constant related to the alternating Lüroth representations of real numbers.