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A260662
Decimal expansion of the generalized Glaisher-Kinkelin constant A(13).
20
1, 2, 2, 2, 9, 4, 4, 2, 5, 1, 8, 0, 8, 1, 3, 3, 8, 7, 2, 6, 4, 7, 8, 9, 9, 9, 6, 0, 7, 2, 7, 7, 1, 7, 9, 8, 8, 5, 6, 1, 2, 6, 5, 8, 0, 3, 1, 2, 9, 5, 3, 2, 9, 5, 0, 1, 0, 8, 3, 7, 2, 8, 1, 0, 3, 4, 4, 6, 0, 6, 4, 2, 2, 7, 6, 8, 6, 6, 2, 0, 3, 0, 3, 0, 0, 1, 2, 6, 4, 2, 6, 9, 2, 1, 7, 5, 1, 1, 4, 2, 6, 1, 2, 4, 4, 9, 1, 8, 3, 6, 0, 0, 2, 0, 9
OFFSET
1,2
COMMENTS
Also known as the thirteenth 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(13) = exp((1/14)*HarmonicNumber(13)*Bernoulli(14) - RiemannZeta'(-13)).
A(13) = exp((B(14)/14)*(EulerGamma + Log(2*Pi) - (zeta'(14)/zeta(14)))).
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^14-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(14)/14 = 1/12 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024
EXAMPLE
1.2229442518081338726478999607277179885...
MATHEMATICA
N[Exp[(1/14)*HarmonicNumber[13]*BernoulliB[14] - Zeta'[-13]], 100]
Exp[N[(BernoulliB[14]/14)*(EulerGamma + Log[2*Pi] - Zeta'[14]/Zeta[14]), 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: A022459 A060804 A086364 * A228044 A171529 A260324
KEYWORD
nonn,cons,easy
AUTHOR
G. C. Greubel, Nov 13 2015
STATUS
approved