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A266556
Decimal expansion of the generalized Glaisher-Kinkelin constant A(9).
20
1, 0, 1, 8, 4, 6, 9, 9, 2, 9, 9, 2, 0, 9, 9, 2, 9, 1, 2, 1, 7, 0, 6, 5, 9, 0, 4, 9, 3, 7, 6, 6, 7, 2, 1, 7, 2, 3, 0, 8, 6, 1, 0, 1, 9, 0, 5, 6, 4, 0, 7, 4, 9, 2, 0, 3, 8, 0, 0, 7, 0, 5, 7, 3, 6, 7, 5, 4, 7, 6, 1, 9, 4, 9, 4
OFFSET
1,4
COMMENTS
Also known as the 9th Bendersky constant.
LINKS
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(9) = exp(H(9)*B(10)/10 - zeta'(-9)) = exp((B(10)/10)*(EulerGamma + log(2*Pi) - (zeta'(10)/zeta(10)))).
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^10-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(10)/10 = 1/132 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024
EXAMPLE
1.018469929920992912170659049376672172308610190564074920380...
MATHEMATICA
Exp[N[(BernoulliB[10]/10)*(EulerGamma + Log[2*Pi] - Zeta'[10]/Zeta[10]), 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)), A266566 (A(19)), A266567 (A(20)).
Sequence in context: A020842 A255987 A372858 * A199434 A253073 A090325
KEYWORD
nonn,cons
AUTHOR
G. C. Greubel, Dec 31 2015
STATUS
approved