A266272 -id:A266272 - cp's OEIS frontend

A266564 Decimal expansion of the generalized Glaisher-Kinkelin constant A(17).

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

Offset: 4

G. C. Greubel, Dec 31 2015

Also known as the 17th Bendersky constant.

			1596.53508575803855385145523662044194533166110061350444341....

G. C. Greubel, Table of n, a(n) for n = 4..2003
Victor S. Adamchik, Polygamma functions of negative order, Journal of Computational and Applied Mathematics, Vol. 100, No. 2 (1998), pp. 191-199.
L. Bendersky, Sur la fonction gamma généralisée, Acta Mathematica , Vol. 61 (1933), pp. 263-322; alternative link.
Robert A. Van Gorder, Glaisher-type products over the primes, International Journal of Number Theory, Vol. 8, No. 2 (2012), pp. 543-550.
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant.

Cf. A019727 (A(0)), A074962 (A(1)), A243262 (A(2)), A243263 (A(3)), A243264 (A(4)), A243265 (A(5)), A266553 (A(6)), A266554 (A(7)), A266555 (A(8)), A266556 (A(9)), A266557 (A(10)), A266558 (A(11)), A266559 (A(12)), A260662 (A(13)), A266560 (A(14)), A266562 (A(15)), A266563 (A(16)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).

Cf. A001620, A013676, A266272, A027641, A027642, A001008, A002805.

Exp[N[(BernoulliB[18]/18)*(EulerGamma + Log[2*Pi] - Zeta'[18]/Zeta[18]), 200]]

A(k) = exp(H(k)*B(k+1)/(k+1) - zeta'(-k)), where B(k) is the k-th Bernoulli number, H(k) the k-th harmonic number, and zeta'(x) is the derivative of the Riemann zeta function.
A(17) = exp(H(17)*B(18)/18 - zeta'(-17)) = exp((B(18)/18)*(EulerGamma + log(2*Pi) - zeta'(18)/zeta(18))).
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^18-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(18)/18 = 43867/14364 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024

A266262 Decimal expansion of zeta'(-11) (the derivative of Riemann's zeta function at -11) (negated).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 25 2015

			-0.012752984479966656113522525488725798156238937049874292793246366661...

G. C. Greubel, Table of n, a(n) for n = 0..1500

Cf. A075700 (zeta'(0)), A084448 (zeta'(-1)), A240966 (zeta'(-2)), A259068 (zeta'(-3)), A259070 (zeta'(-5)), A259071 (zeta'(-6)), A259072 (zeta'(-7)), A259073 (zeta'(-8)), A266260 (zeta'(-9)), A266261 (zeta'(-10)), A266263 (zeta'(-12)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

Join[{0}, RealDigits[Zeta'[-11], 10, 100] // First]

zeta'(-n) = HarmonicNumber(n)*BernoulliB(n+1)/(n+1) - log(A(n)), where A(n) is the n-th Bendersky constant.

zeta'(-11) = - 57844301/908107200 - log(A(11)).

Keyword cons added by Rick L. Shepherd, May 29 2016

A260660 Decimal expansion of zeta'(-13) (the derivative of Riemann's zeta function at -13).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Nov 13 2015

			0.06374987374457688028603868147333505564882735...

G. C. Greubel, Table of n, a(n) for n = 0..1500

Cf. A075700 (zeta'(0)), A084448 (zeta'(-1)), A240966 (zeta'(-2)), A259068 (zeta'(-3)), A259069 (zeta'(-4)), A259070 (zeta'(-5)), A259071 (zeta'(-6)), A259072 (zeta'(-7)), A259073 (zeta'(-8)), A266260 (zeta'(-9)), A266261 (zeta'(-10)), A266262 (zeta'(-11)), A266263 (zeta'(-12)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

N[Zeta'[-13]]
Join[{0}, RealDigits[Zeta'[-13], 10, 1500] // First]

PARI
```
zeta'(-13) \\ Altug Alkan, Nov 13 2015
```

zeta'(-n) = (BernoulliB(n+1)*HarmonicNumber(n))/(n+1) - log(A(n)), where A(n) is the n-th Bendersky constant, that is the n-th generalized Glaisher constant.
zeta'(-13) = (1145993/4324320) - log(A(13)).
zeta'(-13) = 1145993/4324320 - gamma/12 - log(2*Pi)/12 + 6081075*Zeta'(14) / (8*Pi^14), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 05 2015

A266260 Decimal expansion of zeta'(-9) (the derivative of Riemann's zeta function at -9).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 25 2015

			0.0031301453197885727549257682907854467026693658654815.....

G. C. Greubel, Table of n, a(n) for n = 0..1501

Cf. A075700 (zeta'(0)), A084448 (zeta'(-1)), A240966 (zeta'(-2)), A259068 (zeta'(-3)), A259069 (zeta'(-4)), A259070 (zeta'(-5)), A259071 (zeta'(-6)), A259072 (zeta'(-7)), A259073 (zeta'(-8)), A266261 (zeta'(-10)), A266262 (zeta'(-11)), A266263 (zeta'(-12)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

Join[{0, 0}, RealDigits[Zeta'[-9], 10, 100] // First]
N[Zeta'[-9], 100]

zeta'(-n) = HarmonicNumber(n)*BernoulliB(n+1)/(n+1) - log(A(n)), where A(n) is the n-th Bendersky constant.

zeta'(-9) = 7129/332640 - log(A(9)).

A266270 Decimal expansion of zeta'(-15) (the derivative of Riemann's zeta function at -15).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 25 2015

			-0.400319302807725593843580317520320367201261286266232944284106942....

G. C. Greubel, Table of n, a(n) for n = 0..1500

Cf. A075700 (zeta'(0)), A084448 (zeta'(-1)), A240966 (zeta'(-2)), A259068 (zeta'(-3)), A259069 (zeta'(-4)), A259070 (zeta'(-5)), A259071 (zeta'(-6)), A259072 (zeta'(-7)), A259073 (zeta'(-8)), A266260 (zeta'(-9)), A266261 (zeta'(-10)), A266262 (zeta'(-11)), A266263 (zeta'(-12)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

Mathematica
```
RealDigits[N[Zeta'[-15], 100]]
```

zeta'(-n) = (BernoulliB(n+1)*HarmonicNumber(n))/(n+1) - log(A(n)), where A(n) is the n-th Bendersky constant.

zeta'(-15) = -4325053069/2940537600 - log(A(15)).

A266261 Decimal expansion of zeta'(-10) (the derivative of Riemann's zeta function at -10).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 25 2015

			-0.0189299263381403742289805022903467952319852580951695558

G. C. Greubel, Table of n, a(n) for n = 0..1500

Cf. A075700 (zeta'(0)), A084448 (zeta'(-1)), A240966 (zeta'(-2)), A259068 (zeta'(-3)), A259070 (zeta'(-5)), A259071 (zeta'(-6)), A259072 (zeta'(-7)), A259073 (zeta'(-8)), A266260 (zeta'(-9)), A266262 (zeta'(-11)), A266263 (zeta'(-12)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

Join[{0}, RealDigits[-(5/264)*(Zeta[11]/Zeta[10]), 10, 100] // First]

zeta'(-10) = -14175*zeta(11)/(8*Pi^10) = log(A(10)).

Equals -(5/264)*(zeta(11)/zeta(10)).

A266263 Decimal expansion of zeta'(-12) (the derivative of Riemann's zeta function at -12).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 25 2015

			0.06327058334146300059518230123430776751141818475323637667956594567...

G. C. Greubel, Table of n, a(n) for n = 0..1500

Cf. A075700 (zeta'(0)), A084448 (zeta'(-1)), A240966 (zeta'(-2)), A259068 (zeta'(-3)), A259069 (zeta'(-4)), A259070 (zeta'(-5)), A259071 (zeta'(-6)), A259072 (zeta'(-7)), A259073 (zeta'(-8)), A266260 (zeta'(-9)), A266261 (zeta'(-10)), A266262 (zeta'(-11)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

Join[{0}, RealDigits[(691/10920)*(Zeta[13]/Zeta[12]), 10, 100] // First]

zeta'(-12) = (-467775*Zeta(13))/(8*Pi^12) = - log(A(12)).

Equals (691/10920)*(zeta(13)/zeta(12)).

A266264 Decimal expansion of zeta'(-14) (the derivative of Riemann's zeta function at -14).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 25 2015

			-0.29165772474387352032122400307025066697026303853309083214990....

G. C. Greubel, Table of n, a(n) for n = 0..1500

Cf. A075700 (zeta'(0)), A084448 (zeta'(-1)), A240966 (zeta'(-2)), A259068 (zeta'(-3)), A259069 (zeta'(-4)), A259070 (zeta'(-5)), A259071 (zeta'(-6)), A259072 (zeta'(-7)), A259073 (zeta'(-8)), A266260 (zeta'(-9)), A266261 (zeta'(-10)), A266262 (zeta'(-11)), A266263 (zeta'(-12)), A260660 (zeta'(-13)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

Mathematica
```
RealDigits[N[Zeta'[-14], 100]]
```

zeta'(-14) = - (42567525*zeta(15))/(16*Pi^14) = - log(A(14)).

Equals -(7/24)*(zeta(15)/zeta(14)).

A266271 Decimal expansion of zeta'(-16) (the derivative of Riemann's zeta function at -16).

Original entry on oeis.org

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

Offset: 1

G. C. Greubel, Dec 25 2015

			1.7730256608990963962477873441892944813554198276469991771639173077.....

G. C. Greubel, Table of n, a(n) for n = 1..1500

Cf. A075700 (zeta'(0)), A084448 (zeta'(-1)), A240966 (zeta'(-2)), A259068 (zeta'(-3)), A259069 (zeta'(-4)), A259070 (zeta'(-5)), A259071 (zeta'(-6)), A259072 (zeta'(-7)), A259073 (zeta'(-8)), A266260 (zeta'(-9)), A266261 (zeta'(-10)), A266262 (zeta'(-11)), A266263 (zeta'(-12)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

Mathematica
```
RealDigits[N[Zeta'[-16], 100]]
```

zeta'(-16) = (638512875*zeta(17))/(4*Pi^16) = - log(A(16)).

Equals (3617/2040)*(zeta(17)/zeta(16)).

Offset corrected by Rick L. Shepherd, May 21 2016

A266273 Decimal expansion of zeta'(-18) (the derivative of Riemann's zeta function at -18) (negated).

Original entry on oeis.org

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

Offset: 2

G. C. Greubel, Dec 25 2015

			-13.74276825021405443522056419051855107309537215770498560....

G. C. Greubel, Table of n, a(n) for n = 2..1500

Cf. A075700 (zeta'(0)), A084448 (zeta'(-1)), A240966 (zeta'(-2)), A259068 (zeta'(-3)), A259069 (zeta'(-4)), A259070 (zeta'(-5)), A259071 (zeta'(-6)), A259072 (zeta'(-7)), A259073 (zeta'(-8)), A266260 (zeta'(-9)), A266261 (zeta'(-10)), A266262 (zeta'(-11)), A266263 (zeta'(-12)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

Mathematica
```
RealDigits[N[Zeta'[-18], 100]]
```

zeta'(-18) = -(97692469875*zeta(19))/(8*Pi^18) = - log(A(18)).

Equals -(43867/3192)*(zeta(19)/zeta(18)).

Offset corrected by Rick L. Shepherd, May 30 2016

cp's OEIS Frontend

A266564 Decimal expansion of the generalized Glaisher-Kinkelin constant A(17).

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A266262 Decimal expansion of zeta'(-11) (the derivative of Riemann's zeta function at -11) (negated).

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A260660 Decimal expansion of zeta'(-13) (the derivative of Riemann's zeta function at -13).

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A266260 Decimal expansion of zeta'(-9) (the derivative of Riemann's zeta function at -9).

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A266270 Decimal expansion of zeta'(-15) (the derivative of Riemann's zeta function at -15).

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A266261 Decimal expansion of zeta'(-10) (the derivative of Riemann's zeta function at -10).

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A266263 Decimal expansion of zeta'(-12) (the derivative of Riemann's zeta function at -12).

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A266264 Decimal expansion of zeta'(-14) (the derivative of Riemann's zeta function at -14).

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