A243265 Decimal expansion of the generalized Glaisher-Kinkelin constant A(5).

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

Offset

1,4

Comments

Also known as the 5th Bendersky constant.

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.

Links

G. C. Greubel, Table of n, a(n) for n = 1..2004

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 MathWorld, Glaisher-Kinkelin Constant.

Formula

A(k) = exp(B(k+1)/(k+1)*H(k)-zeta'(-k)), where B(k) is the k-th Bernoulli number and H(k) the k-th harmonic number.

A(5) = exp(137/15120-zeta'(-5)).

Equals exp(gamma/252 - 15*Zeta'(6)/(4*Pi^6)) * (2*Pi)^(1/252), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 25 2015

Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^6-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(6)/6 = 1/252 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024

Example

1.00968038728586616112008919046263...

Mathematica

RealDigits[Exp[137/15120-Zeta'[-5]], 10, 103] // First
RealDigits[Exp[N[(BernoulliB[6]/6)*(EulerGamma + Log[2*Pi] - Zeta'[6]/Zeta[6]), 200]]]//First (* G. C. Greubel, Dec 31 2015 *)

Crossrefs

Cf. A001620, A255344, A259070.

Cf. A019727, A074962, A243262, A243263, A243264, A266553, A266554, A266555, A266556, A266557, A266558, A266559, A260662, A266560, A266562, A266563, A266564, A266565, A266566, A266567.

Sequence in context: A019643 A011012 A157989 * A248472 A306553 A011194

Adjacent sequences: A243262 A243263 A243264 * A243266 A243267 A243268

Keyword

nonn,cons

Author

Jean-François Alcover, Jun 02 2014

Status

approved