OFFSET
4,2
COMMENTS
Also known as the 17th Bendersky constant.
LINKS
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.
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(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
EXAMPLE
1596.53508575803855385145523662044194533166110061350444341....
MATHEMATICA
Exp[N[(BernoulliB[18]/18)*(EulerGamma + Log[2*Pi] - Zeta'[18]/Zeta[18]), 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)), A266563 (A(16)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).
KEYWORD
nonn,cons
AUTHOR
G. C. Greubel, Dec 31 2015
STATUS
approved