A260662 - OEIS

%I #42 Feb 08 2024 22:31:33

%S 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,

%T 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,

%U 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

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

%C Also known as the thirteenth Bendersky constant.

%H G. C. Greubel, <a href="/A260662/b260662.txt">Table of n, a(n) for n = 1..2001</a>

%H Victor S. Adamchik, <a href="https://doi.org/10.1016/S0377-0427(98)00192-7">Polygamma functions of negative order</a>, Journal of Computational and Applied Mathematics, Vol. 100, No. 2 (1998), pp. 191-199.

%H L. Bendersky, <a href="https://doi.org/10.1007/BF02547794">Sur la fonction gamma généralisée</a>, Acta Mathematica , Vol. 61 (1933), pp. 263-322; <a href="https://projecteuclid.org/journals/acta-mathematica/volume-61/issue-none/Sur-la-fonction-gamma-g%C3%A9n%C3%A9ralis%C3%A9e/10.1007/BF02547794.full">alternative link</a>.

%H Robert A. Van Gorder, <a href="https://doi.org/10.1142/S1793042112500297">Glaisher-type products over the primes</a>, International Journal of Number Theory, Vol. 8, No. 2 (2012), pp. 543-550.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Glaisher-KinkelinConstant.html">Glaisher-Kinkelin Constant</a>.

%F 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.

%F A(13) = exp((1/14)*HarmonicNumber(13)*Bernoulli(14) - RiemannZeta'(-13)).

%F A(13) = exp((B(14)/14)*(EulerGamma + Log(2*Pi) - (zeta'(14)/zeta(14)))).

%F 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

%e 1.2229442518081338726478999607277179885...

%t N[Exp[(1/14)*HarmonicNumber[13]*BernoulliB[14] - Zeta'[-13]], 100]

%t Exp[N[(BernoulliB[14]/14)*(EulerGamma + Log[2*Pi] - Zeta'[14]/Zeta[14]), 200]]

%Y 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)).

%Y Cf. A001620, A260660, A260663.

%K nonn,cons,easy

%O 1,2

%A _G. C. Greubel_, Nov 13 2015