A243263 -id:A243263 - cp's OEIS frontend

A074962 Decimal expansion of Glaisher-Kinkelin constant A.

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

Offset: 1

Benoit Cloitre, Oct 05 2002

Arises in expressions such as A002109(n) = 1^1*2^2*3^3*...*n^n which is asymptotic to A*n^(n^2/2 + n/2 + 1/12)*exp(-n^2/4). See A002109 for more references and links.

Named after the English mathematician and astronomer James Whitbread Lee Glaisher (1848-1928) and the Swiss mathematician Hermann Kinkelin (1832-1913). - Amiram Eldar, Jun 15 2021

			1.2824271291006226368753425688697917277676889273250011920637400217404...

Steven R. Finch, Mathematical constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, p. 135.
Konrad Knopp, Theory and applications of infinite series, Dover, p. 555.

Gheorghe Coserea, Table of n, a(n) for n = 1..10010
L. Almodovar, V. H. Moll, H. Quand, Infinite products arising in paperfolding, JIS 19 (2016) # 16.5.1 eq. (13)
Chao-Ping Chen and Long Lin, Asymptotic expansions related to Glaisher-Kinkelin constant based on the Bell polynomials, Journal of Number Theory, Vol. 133 (2013), pp. 2699-2705.
Ovidiu Furdui, proposer, Problem 11494, Amer. Math. Monthly, Vol. 118, No. 9 (2011), 850-852.
J. W. L. Glaisher, On the Product 1^1.2^2.3^3...n^n, The Messenger of Mathematics, Vol. 7 (1878), pp. 43-47.
Antonio Gracia Llorente, A Simple Limit-Product Formula for Glaisher’s Constant, OSF Preprint, 2025.
Jesús Guillera and Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J., Vol. 16 (2008), pp. 247-270; see Examples 5.2, 5.7, 5.11.
Fredrik Johansson et al., mpmath, Mathematical constants (Mpmath).
Fredrik Johansson et al., mpmath, Glaisher's constant to 20,000 digits.
Hermann Kinkelin, Über eine mit der Gammafunction verwandte Transcendente und deren Anwendung auf die Integralrechnung, Journal für die reine und angewandte Mathematik, Vol. 57 (1860), pp. 122-138.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl., Vol. 332, No. 1 (2007), pp. 292-314; see Section 5.
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.
Wikipedia, Glaisher-Kinkelin constant.

Cf. A001620, A243262, A243263, A243264, A243265.

Cf. A000178, A002109, A051675, A255321, A255323, A255344.

evalf(limit(product(k^k,k=1..n)/(n^(n^2/2+n/2+1/12)*exp(-n^2/4)),n=infinity),120); # Vaclav Kotesovec, Oct 23 2014

RealDigits[Glaisher, 10, 111][[1]] (* Robert G. Wilson v *)

x=10^(-100); exp(1/12-(zeta(-1+x)-zeta(-1))/x)

exp(1/12-zeta'(-1)) \\ Charles R Greathouse IV, Dec 12 2013

A = 2^(1/36)*Pi^(1/6)*exp(1/3*(-Gamma/4 + s(2)/3 - s(3)/4 + ...)) where s(k) denotes Sum_{n>=0} 1/(2n+1)^k.
Closed expressions for A are exp(-zeta'(2)/2/Pi^2 + log(2*Pi)/12 + Gamma/12) or exp(1/12-zeta'(-1)).
Equals (2*Pi)^(1/4) / limit_{n->oo} Product_{k=1..n} Gamma(k/n)^(k/n^2). - Vaclav Kotesovec, Dec 02 2023
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^4-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(2)/2 = 1/12 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024
Equals e^(-1/4 + Integral_{x=1..2} x*log(sqrt(2*Pi)) - B_2(x) + x^2*Psi(x)/2 dx), where B_2(x) is the second Bernoulli polynomial and Psi(x) is the digamma function. - Andrea Pinos, Apr 16 2024
Equals exp(1/12 - 2*Integral_{x=0..oo} x*log(x)/(exp(2*Pi*x) - 1) dx) = exp(1/3 + 7*log(2)/36 - log(Pi)/6 + (2/3)*Integral_{x=0..1/2} log(Gamma(x+1)) dx) (see Finch). - Stefano Spezia, Dec 01 2024
From Antonio Graciá Llorente, May 03 2025: (Start)
Equals lim_{n->oo} (2^(13/3)*n)^(1/12) * Product_{k=1..n} (1 - 1/(2*k+1)^2)^((2*k+1)/6).
Equals lim_{n->oo} (24*n^2)^(1/24) * Product_{prime p<=n} (p^(1 - p/(p^2-1)) / sqrt(p^2-1))^(1/12). (End)

More terms from Sascha Kurz, Feb 03 2003

A243262 Decimal expansion of the generalized Glaisher-Kinkelin constant A(2).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 02 2014

Also known as the second Bendersky constant.

This is likely the same as the constant B considered in section 3 of the Choi and Srivastava link. - R. J. Mathar, Oct 03 2016

			1.03091675219739211419331309646694229...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.

G. C. Greubel, Table of n, a(n) for n = 1..2002
J. Choi and H. M. Srivastava, Certain classes of series involving the zeta function, J. Math. Annal. Applic. 231 (1999) 91-117.
K. Kimoto, N. Kurokawa, C. Sonoki, M. Wakayama, Some examples of generalized zeta regularized products, Kodai Math. J. 27 (2004), 321-335.
Tobias Kyrion, A closed-form expression for zeta(3), arXiv:2008.05573 [math.GM], 2020.
Eric Weisstein's MathWorld, Glaisher-Kinkelin Constant

Cf. A051675, A055462, A240966, A255269.
Cf. A019727, A074962, A243263, A243264, A243265, A266553, A266554, A266555, A266556, A266557, A266558, A266559, A260662, A266560, A266562, A266563, A266564, A266565, A266566, A266567.
Cf. A002117 (zeta(3)).

RealDigits[Exp[Zeta[3]/(4*Pi^2)], 10, 99] // First
RealDigits[Exp[N[(BernoulliB[2]/4)*(Zeta[3]/Zeta[2]), 200]]]//First (* G. C. Greubel, Dec 31 2015 *)

exp(zeta(3)/(4*Pi^2)) \\ Felix Fröhlich, Jun 27 2019

A(k) = exp(B(k+1)/(k+1)*H(k)-zeta'(-k)), where B(k) is the k-th Bernoulli number and H(k) the k-th harmonic number.
A(0) = sqrt(2*Pi) (A019727),
A(1) = A = Glaisher-Kinkelin constant (A074962),
A(2) = exp(-zeta'(-2)) = exp(zeta(3)/(4*Pi^2)).
Equals exp(-A240966). - Vaclav Kotesovec, Feb 22 2015

A243264 Decimal expansion of the generalized Glaisher-Kinkelin constant A(4).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 02 2014

Also known as the 4th Bendersky constant.

			0.9920479745250402600134369776254433567369...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.

G. C. Greubel, Table of n, a(n) for n = 0..2002
Eric Weisstein's MathWorld, Glaisher-Kinkelin Constant.

Cf. A255323, A259069.

Cf. A019727, A074962, A243262, A243263, A243265, A266553, A266554, A266555, A266556, A266557, A266558, A266559, A260662, A266560, A266562, A266563, A266564, A266565, A266566, A266567.

RealDigits[Exp[-3*Zeta[5]/(4*Pi^4)], 10, 98] // First
RealDigits[Exp[N[(BernoulliB[4]/4)*(Zeta[5]/Zeta[4]), 100]]] // First (* G. C. Greubel, Dec 31 2015 *)

exp(-3*zeta(5)/(4*Pi^4)) \\ Stefano Spezia, Dec 01 2024

A(k) = exp(B(k+1)/(k+1)*H(k)-zeta'(-k)), where B(k) is the k-th Bernoulli number and H(k) the k-th harmonic number.
A(4) = exp(-zeta'(-4)) = exp(-3*zeta(5)/(4*Pi^4)).
A(4) = exp((B(4)/4)*(zeta(5)/zeta(4))). - G. C. Greubel, Dec 31 2015

A243265 Decimal expansion of the generalized Glaisher-Kinkelin constant A(5).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 02 2014

Also known as the 5th Bendersky constant.

			1.00968038728586616112008919046263...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.

G. C. Greubel, Table of n, a(n) for n = 1..2004
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 MathWorld, Glaisher-Kinkelin Constant.

Cf. A001620, A255344, A259070.

Cf. A019727, A074962, A243262, A243263, A243264, A266553, A266554, A266555, A266556, A266557, A266558, A266559, A260662, A266560, A266562, A266563, A266564, A266565, A266566, A266567.

RealDigits[Exp[137/15120-Zeta'[-5]], 10, 103] // First
RealDigits[Exp[N[(BernoulliB[6]/6)*(EulerGamma + Log[2*Pi] - Zeta'[6]/Zeta[6]), 200]]]//First (* G. C. Greubel, Dec 31 2015 *)

exp(137/15120-zeta'(-5)) \\ Stefano Spezia, Dec 01 2024

A(k) = exp(B(k+1)/(k+1)*H(k)-zeta'(-k)), where B(k) is the k-th Bernoulli number and H(k) the k-th harmonic number.
A(5) = exp(137/15120-zeta'(-5)).
Equals exp(gamma/252 - 15*Zeta'(6)/(4*Pi^6)) * (2*Pi)^(1/252), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 25 2015
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^6-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(6)/6 = 1/252 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024

A259068 Decimal expansion of zeta'(-3) (the derivative of Riemann's zeta function at -3).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 18 2015

			0.0053785763577743011444169742104138428956644397422955070594470232233245...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15.1 Generalized Glaisher constants, p. 136-137.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's MathWorld, Riemann Zeta Function.
Wikipedia, Riemann Zeta Function
Index entries for constants related to zeta

Cf. A000335, A000391, A000417, A000428, A023872, A057527, A057528, A255050, A255052, A258350, A258351, A258352, A260404.

Join[{0, 0}, RealDigits[Zeta'[-3], 10, 105] // First]

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'(-3) = -11/720 - log(A(3)), where A(3) is A243263.
Equals -11/720 + (gamma + log(2*Pi))/120 - 3*Zeta'(4)/(4*Pi^4), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 24 2015

A260662 Decimal expansion of the generalized Glaisher-Kinkelin constant A(13).

Original entry on oeis.org

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

Offset: 1

G. C. Greubel, Nov 13 2015

Also known as the thirteenth Bendersky constant.

			1.2229442518081338726478999607277179885...

G. C. Greubel, Table of n, a(n) for n = 1..2001
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)), A266564 (A(17)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).

Cf. A001620, A260660, A260663.

N[Exp[(1/14)*HarmonicNumber[13]*BernoulliB[14] - Zeta'[-13]], 100]
Exp[N[(BernoulliB[14]/14)*(EulerGamma + Log[2*Pi] - Zeta'[14]/Zeta[14]), 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(13) = exp((1/14)*HarmonicNumber(13)*Bernoulli(14) - RiemannZeta'(-13)).
A(13) = exp((B(14)/14)*(EulerGamma + Log(2*Pi) - (zeta'(14)/zeta(14)))).
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^14-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(14)/14 = 1/12 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024

A266554 Decimal expansion of the generalized Glaisher-Kinkelin constant A(7).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 31 2015

Also known as the 7th Bendersky constant.

			0.9899756533334170941753964830588692002082471514307453051285538624....

G. C. Greubel, Table of n, a(n) for n = 0..2000
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)), 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)), A266564 (A(17)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).

Cf. A001620, A013666, A259072, A027641, A027642, A001008, A002805.

Exp[N[(BernoulliB[8]/8)*(EulerGamma + Log[2*Pi] - Zeta'[8]/Zeta[8]), 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(7) = exp(H(7)*B(8)/8 - zeta'(-7)) = exp((B(8)/8)*(EulerGamma + log(2*Pi) - (zeta'(8)/zeta(8)))).
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^8-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(8)/8 = -1/240 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024

A266556 Decimal expansion of the generalized Glaisher-Kinkelin constant A(9).

Original entry on oeis.org

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

Offset: 1

G. C. Greubel, Dec 31 2015

Also known as the 9th Bendersky constant.

			1.018469929920992912170659049376672172308610190564074920380...

G. C. Greubel, Table of n, a(n) for n = 1..2002
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)), A266564 (A(17)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).

Cf. A001620, A013668, A266260, A027641, A027642, A001008, A002805.

Exp[N[(BernoulliB[10]/10)*(EulerGamma + Log[2*Pi] - Zeta'[10]/Zeta[10]), 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(9) = exp(H(9)*B(10)/10 - zeta'(-9)) = exp((B(10)/10)*(EulerGamma + log(2*Pi) - (zeta'(10)/zeta(10)))).
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^10-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(10)/10 = 1/132 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024

A266558 Decimal expansion of the generalized Glaisher-Kinkelin constant A(11).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 31 2015

Also known as the 11th Bendersky constant.

			0.950331248453288866514233841015331271597566403456173040861088881...

G. C. Greubel, Table of n, a(n) for n = 0..2000
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)), A266559 (A(12)), A260662 (A(13)), A266560 (A(14)), A266562 (A(15)), A266563 (A(16)), A266564 (A(17)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).

Cf. A001620, A013670, A266262, A027641, A027642, A001008, A002805.

Exp[N[(BernoulliB[12]/12)*(EulerGamma + Log[2*Pi] - Zeta'[12]/Zeta[12]), 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(11) = exp(H(11)*B(12)/12 - zeta'(-11)) = exp((B(12)/12)*(EulerGamma + log(2*Pi) - (zeta'(12)/zeta(12)))).
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^12-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(12)/12 = -691/32760 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024

A266562 Decimal expansion of the generalized Glaisher-Kinkelin constant A(15).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 31 2015

Also known as the 15th Bendersky constant.

			0.342830806132816736571711146340672378141726945483236877251076164....

G. C. Greubel, Table of n, a(n) for n = 0..2000
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)), A266563 (A(16)), A266564 (A(17)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).

Cf. A001620, A013674, A266270, A027641, A027642, A001008, A002805.

Exp[N[(BernoulliB[16]/16)*(EulerGamma + Log[2*Pi] - Zeta'[16]/Zeta[16]), 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(15) = exp(H(15)*B(16)/16 - zeta'(-15)) = exp((B(16)/16)*(EulerGamma + log(2*Pi) - zeta'(16)/zeta(16))).
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^16-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(16)/16 = -3617/8160 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024

cp's OEIS Frontend

A074962 Decimal expansion of Glaisher-Kinkelin constant A.

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A243262 Decimal expansion of the generalized Glaisher-Kinkelin constant A(2).

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A243264 Decimal expansion of the generalized Glaisher-Kinkelin constant A(4).

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A243265 Decimal expansion of the generalized Glaisher-Kinkelin constant A(5).

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A259068 Decimal expansion of zeta'(-3) (the derivative of Riemann's zeta function at -3).

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A260662 Decimal expansion of the generalized Glaisher-Kinkelin constant A(13).

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A266554 Decimal expansion of the generalized Glaisher-Kinkelin constant A(7).

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