A266554 Decimal expansion of the generalized Glaisher-Kinkelin constant A(7).

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

Offset

0,1

Comments

Also known as the 7th Bendersky constant.

Links

G. C. Greubel, Table of n, a(n) for n = 0..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(7) = exp(H(7)*B(8)/8 - zeta'(-7)) = exp((B(8)/8)*(EulerGamma + log(2*Pi) - (zeta'(8)/zeta(8)))).

Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^8-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(8)/8 = -1/240 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024

Example

0.9899756533334170941753964830588692002082471514307453051285538624....

Mathematica

Exp[N[(BernoulliB[8]/8)*(EulerGamma + Log[2*Pi] - Zeta'[8]/Zeta[8]), 200]]

Crossrefs

Cf. A019727 (A(0)), A074962 (A(1)), A243262 (A(2)), A243263 (A(3)), A243264 (A(4)), A243265 (A(5)), A266553 (A(6)), 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)).

Cf. A001620, A013666, A259072, A027641, A027642, A001008, A002805.

Sequence in context: A109940 A275106 A010549 * A046267 A343765 A019327

Adjacent sequences: A266551 A266552 A266553 * A266555 A266556 A266557

Keyword

nonn,cons

Author

G. C. Greubel, Dec 31 2015

Status

approved