OFFSET
1,4
COMMENTS
Also known as the 5th Bendersky constant.
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.
LINKS
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.
FORMULA
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
EXAMPLE
1.00968038728586616112008919046263...
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Jun 02 2014
STATUS
approved