A266564 Decimal expansion of the generalized Glaisher-Kinkelin constant A(17).

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

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)).

Cf. A001620, A013676, A266272, A027641, A027642, A001008, A002805.

Sequence in context: A087498 A274633 A238200 * A201589 A198349 A370742

Adjacent sequences: A266561 A266562 A266563 * A266565 A266566 A266567

Keyword

nonn,cons

Author

G. C. Greubel, Dec 31 2015

Status

approved