A259070 -id:A259070 - cp's OEIS frontend

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

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

A266274 Decimal expansion of zeta'(-19) (the derivative of Riemann's zeta function at -19) (negated).

Original entry on oeis.org

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

Offset: 2

G. C. Greubel, Dec 26 2015

			-29.965529831392351939431865297274201791908226109115565915881....

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)), A266273 (zeta'(-18)), A266275 (zeta'(-20)).

Mathematica
```
RealDigits[N[Zeta'[-19], 100]]
```

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

zeta'(-19) = -48069674759189/512143632000 - log(A(19)).

Offset corrected by Rick L. Shepherd, May 30 2016

A266275 Decimal expansion of zeta'(-20) (the derivative of Riemann's zeta function at -20).

Original entry on oeis.org

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

Offset: 3

G. C. Greubel, Dec 26 2015

			132.28099750421251452709821158578551868064800999955031458847450192414...

G. C. Greubel, Table of n, a(n) for n = 3..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)), A266273 (zeta'(-18)), A266274 (zeta'(-19)).

Mathematica
```
RealDigits[N[Zeta'[-20], 100]]
```

zeta'(-20) = (9280784638125*zeta(21))/(8*Pi^20) = - log(A(20)).

Equals (174611/1320)*(zeta(21)/zeta(20)).

Offset corrected by Rick L. Shepherd, May 30 2016

A261506 Decimal expansion of -zeta'(4).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Aug 22 2015

			0.06891126589612537984882936558744082715001637487137...

J. B. Rosser, L. Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. J. Math. 6 (1) (1962) 64-94, Table IV

Cf. A075700 (0), A073002 (2), A244115 (3).

Cf. A084448 (-1), A240966 (-2), A259068 (-3), A259069 (-4), A259070 (-5), A259071 (-6), A259072 (-7), A259073 (-8).

Flatten[{0, RealDigits[-Zeta'[4], 10, 105][[1]]}]

Sum_{n>=1} log(n) / n^4.

A260404 6th level factorials: product of first n 5th level factorials.

Original entry on oeis.org

1, 1, 2, 192, 6115295232, 15436756676507918107049554083840, 18356962141505758798331790171539976807981714702571497465907439808868887035904000000

Offset: 0

Vaclav Kotesovec, Jul 24 2015

nonn

In general for k-th level factorials a(n) = Product_{j=1..n} j^C(n-j+k-1,k-1).

Cf. A000142, A000178, A055462, A057527, A057528.

Table[Product[i^Binomial[n-i+5,5],{i,1,n}],{n,0,10}]

a(n) ~ exp(137/720 - 11*n/16 - 737*n^2/480 - 53*n^3/48 - 421*n^4/1152 - 137*n^5/2400 - 49*n^6/14400 + (3 + n)*(15 + 12*n + 2*n^2)*Zeta(3)/(96*Pi^2) - (3 + n)*Zeta(5) / (32*Pi^4) + (17 + 12*n + 2*n^2)*Zeta'(-3)/24 + Zeta'(-5)/120) * n^(19087/60480 + n + 137*n^2/120 + 5*n^3/8 + 17*n^4/96 + n^5/40 + n^6/720) * (2*Pi)^((n+1)*(n+2)*(n+3)*(n+4)*(n+5)/240) / A^(137/60 + 15*n/4 + 17*n^2/8 + n^3/2 + n^4/24), where Zeta(3) = A002117, Zeta(5) = A013663, Zeta'(-3) = A259068, Zeta'(-5) = A259070 and A = A074962 is the Glaisher-Kinkelin constant.

A255439 Decimal expansion of a constant related to A255360.

Original entry on oeis.org

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

Offset: 2

Vaclav Kotesovec, Feb 24 2015

			11.354954749729782312106630592450215781014...

Cf. A255360, A255504, A255511, A255438.

Cf. A074962, A243262, A243263, A243264, A243265.

Equals limit n->infinity (Product_{k=0..n} (k^5)!) / (n^(80/63 + 5*n/2 - 5*n^2/12 + 25*n^4/12 + 5*n^5/2 + (5*n^6)/6) * (2*Pi)^(n/2) / exp(5*n/2 + 35*n^2/144 + n^5/2 + 11*n^6/36)).

Equals 2^(5/4)*Pi^(5/4)*exp(137/3024 - 5*Zeta'(-5)) * Product_{n>=1} ((n^5)! / stirling(n^5)), where stirling(n^5) = sqrt(2*Pi) * n^(5*n^5 + 5/2) / exp(n^5) is the Stirling approximation of (n^5)! and Zeta'(-5) = A259070. - Vaclav Kotesovec, Apr 20 2016

A271172 Decimal expansion of the logarithm of the generalized Glaisher-Kinkelin constant A(5).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Apr 01 2016

The logarithm of the fifth Bendersky constant.

			0.009633832541045196051551840709680435359814...

G. C. Greubel, Table of n, a(n) for n = 0..2000

Cf. A243265, A259070.

Join[{0, 0}, RealDigits[(BernoulliB[6]/6)*(EulerGamma + Log[2*Pi] - Zeta'[6]/Zeta[6]), 10, 100] // First]

log(A(5)) = (1/6)*HarmonicNumber(5)*Bernoulli(6) - RiemannZeta'(-5).

log(A(5)) = (BernoulliB(6)/6)*(EulerGamma + log(2*Pi) - Zeta'(6)/Zeta(6)).

A271854 Decimal expansion of -zeta'(-1/2), negated derivative of the Riemann zeta function at -1/2.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Apr 23 2016

			zeta'(-1/2) = -0.36085433959994760734742080636395106588485278791863221...

Stanislav Sykora, Table of n, a(n) for n = 0..1000

Values of |zeta'(x)| for various x: A073002 (+2), A075700 (0), A084448 (-1), A114875 (+1/2), A240966 (-2), A244115(+3), A259068 (-3), A259069 (-4), A259070 (-5), A259071 (-6), A259072 (-7), A259073 (-8), A261506 (+4), A266260 (-9), A266261 (-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)), A271521 (i).

RealDigits[N[-Zeta'[-1/2], 106]] [[1]] (* Robert Price, Apr 28 2016 *)

PARI
```
-zeta'(-1/2)
```

cp's OEIS Frontend

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

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A266274 Decimal expansion of zeta'(-19) (the derivative of Riemann's zeta function at -19) (negated).

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A266275 Decimal expansion of zeta'(-20) (the derivative of Riemann's zeta function at -20).

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A261506 Decimal expansion of -zeta'(4).

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A260404 6th level factorials: product of first n 5th level factorials.

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A255439 Decimal expansion of a constant related to A255360.

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A271172 Decimal expansion of the logarithm of the generalized Glaisher-Kinkelin constant A(5).

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Mathematica

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A271854 Decimal expansion of -zeta'(-1/2), negated derivative of the Riemann zeta function at -1/2.

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