A266271 -id:A266271

A266563

Decimal expansion of the generalized Glaisher-Kinkelin constant A(16).

+10
19

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Also known as the 16th Bendersky constant.

LINKS

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

FORMULA

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(16) = exp((B(16)/4)*(zeta(17)/zeta(16))) = exp(-zeta'(-16)).

A(16) = exp(-16! * Zeta(17) / (2^17 * Pi^16)). - Vaclav Kotesovec, Jan 01 2016

EXAMPLE

0.16981839784277560774730955168312711879515291428637735860...

MATHEMATICA

Exp[N[(BernoulliB[16]/4)*(Zeta[17]/Zeta[16]), 200]]

CROSSREFS

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)), A266564 (A(17)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).

Cf. A013674, A013675, A266271, A027641, A027642.

KEYWORD

nonn,cons

AUTHOR

G. C. Greubel, Dec 31 2015

STATUS

approved

A266262

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

+10
16

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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

FORMULA

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)).

EXAMPLE

-0.012752984479966656113522525488725798156238937049874292793246366661...

MATHEMATICA

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

CROSSREFS

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)).

KEYWORD

nonn,cons

AUTHOR

G. C. Greubel, Dec 25 2015

EXTENSIONS

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

STATUS

approved

A260660

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

+10
15

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

FORMULA

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

EXAMPLE

0.06374987374457688028603868147333505564882735...

MATHEMATICA

N[Zeta'[-13]]

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

PROG

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

CROSSREFS

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)).

KEYWORD

nonn,cons

AUTHOR

G. C. Greubel, Nov 13 2015

STATUS

approved

A266260

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

+10
15

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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

FORMULA

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)).

EXAMPLE

0.0031301453197885727549257682907854467026693658654815.....

MATHEMATICA

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

N[Zeta'[-9], 100]

CROSSREFS

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)).

KEYWORD

nonn,cons,easy

AUTHOR

G. C. Greubel, Dec 25 2015

STATUS

approved

A266270

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

+10
15

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

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

FORMULA

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)).

EXAMPLE

-0.400319302807725593843580317520320367201261286266232944284106942....

MATHEMATICA

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

CROSSREFS

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)).

KEYWORD

nonn,cons

AUTHOR

G. C. Greubel, Dec 25 2015

STATUS

approved

A266272

Decimal expansion of zeta'(-17) (the derivative of Riemann's zeta function at -17).

+10
15

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

FORMULA

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

zeta'(-17) = 1848652896341/175991175360 - log(A(17)).

EXAMPLE

3.1286453321241578756844526391533305482263390775654797424916577061....

MATHEMATICA

RealDigits[N[Zeta'[-17], 100]]

CROSSREFS

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)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

KEYWORD

nonn,cons

AUTHOR

G. C. Greubel, Dec 25 2015

EXTENSIONS

Offset corrected by Rick L. Shepherd, May 21 2016

STATUS

approved

A266261

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

+10
14

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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

FORMULA

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

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

EXAMPLE

-0.0189299263381403742289805022903467952319852580951695558

MATHEMATICA

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

CROSSREFS

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)).

KEYWORD

nonn,cons

AUTHOR

G. C. Greubel, Dec 25 2015

STATUS

approved

A266263

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

+10
14

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

FORMULA

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

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

EXAMPLE

0.06327058334146300059518230123430776751141818475323637667956594567...

MATHEMATICA

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

CROSSREFS

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)).

KEYWORD

nonn,cons

AUTHOR

G. C. Greubel, Dec 25 2015

STATUS

approved

A266264

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

+10
14

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

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

FORMULA

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

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

EXAMPLE

-0.29165772474387352032122400307025066697026303853309083214990....

MATHEMATICA

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

CROSSREFS

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)).

KEYWORD

nonn,cons

AUTHOR

G. C. Greubel, Dec 25 2015

STATUS

approved

A266273

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

+10
14

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

2,2

LINKS

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

FORMULA

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

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

EXAMPLE

-13.74276825021405443522056419051855107309537215770498560....

MATHEMATICA

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

CROSSREFS

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)).

KEYWORD

nonn,cons

AUTHOR

G. C. Greubel, Dec 25 2015

EXTENSIONS

Offset corrected by Rick L. Shepherd, May 30 2016

STATUS

approved