A266566 Decimal expansion of the generalized Glaisher-Kinkelin constant A(19).

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

Offset

-28,2

Comments

Also known as the 19th Bendersky constant.

Links

G. C. Greubel, Table of n, a(n) for n = -28..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.

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(19) = exp(H(19)*B(20)/20 - zeta'(-19)) = exp((B(20)/20)*(EulerGamma + log(2*Pi) - (zeta'(20)/zeta(20))).

Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^20-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(20)/20 = -174611/6600 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024

Example

1.78275427694671411881767097743544556956081837015720653244894...*10^(-28)

Mathematica

Exp[N[(BernoulliB[20]/20)*(EulerGamma + Log[2*Pi] - Zeta'[20]/Zeta[20]), 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)), A266564 (A(17)), A266565 (A(18)), A266567 (A(20)).

Cf. A001620.

Sequence in context: A011103 A342486 A245758 * A309523 A153622 A257576

Adjacent sequences: A266563 A266564 A266565 * A266567 A266568 A266569

Keyword

nonn,cons

Author

G. C. Greubel, Dec 31 2015

Status

approved