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%I #88 Mar 27 2024 20:10:55
%S 1,2,5,7,7,4,6,8,8,6,9,4,4,3,6,9,6,3,0,0,0,9,8,9,9,8,3,0,4,9,5,8,8,1,
%T 5,2,8,5,1,1,5,4,0,8,9,0,5,0,8,8,8,4,8,6,8,9,7,7,5,4,0,8,3,3,5,2,2,5,
%U 4,9,9,9,4,8,9,3,7,4,4,9,3,4,9,7,0,7,9,0,4,7,3,1,5,0,1,9,0,9,7,8,2,4,5,4,8
%N Decimal expansion of the constant Sum_{k>=1} log(k + 1) / (k * (k + 1)).
%C Given A131385(n) = Product_{k=1..n} floor((n+k)/k), then limit A131385(n+1)/A131385(n) = exp(c), where c = this constant. - _Paul D. Hanna_, Nov 26 2012
%C Closely related to A085361 (the exponent in the definition of A085291). - _Yuriy Sibirmovsky_, Sep 04 2016
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, page 62. [_Jean-François Alcover_, Mar 21 2013]
%H G. C. Greubel, <a href="/A131688/b131688.txt">Table of n, a(n) for n = 1..1000</a>
%H Khristo N. Boyadzhiev, <a href="https://arxiv.org/abs/1903.11141">A special constant and series with zeta values and harmonic numbers</a>, arXiv:1903.11141 [math.NT], 2019.
%H Mark W. Coffey, <a href="http://www.arXiv.org/abs/0706.0345">Series of zeta values, the Stieltjes constants and a sum S_gamma(n)</a>, arXiv:math-ph/0706.0345, 2007-2009, eq (38a).
%H Paul Erdős, S. W. Graham, Aleksandar Ivic and Carl Pomerance, <a href="http://math.dartmouth.edu/~carlp/factorial.pdf">On the number of divisors of n!</a>, Analytic Number Theory, Volume 138, Progress in Mathematics pp 337-355.
%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 538.
%H Sofia Kalpazidou, <a href="http://dx.doi.org/10.1016/0022-314X(88)90099-6">Khintchine's constant for Lüroth representation</a>, Journal of Number Theory, Volume 29, Issue 2, June 1988, Pages 196-205.
%F Equals Sum_{s>=1} (-1)^(s+1)*zeta(s+1)/s.
%F Equals Sum_{k>=1} -zeta'(1 + k), where Zeta' is the derivative of the Riemann zeta function. - _Vladimir Reshetnikov_, Dec 28 2008
%F Equals Sum_{s>=1} log(1+1/s)/s. - _Jean-François Alcover_, Mar 26 2013
%F Equals Integral_{t=0..1} H(t)/t dt. Compare to A001620 = Integral_{t=0..1} H(t) dt. Where H(t) are generalized harmonic numbers. - _Yuriy Sibirmovsky_, Sep 04 2016
%F Equals lim_{n->oo} log(d(n!))*log(n)/n, where d(n) is the number of divisors of n (A000005) (Erdős et al., 1996). - _Amiram Eldar_, Nov 07 2020
%e 1.257746886944369630009899830495881528511540890508884868977540833522...
%p evalf(sum((-1)^(n+1)*Zeta(n+1)/n, n=1..infinity), 120); # _Vaclav Kotesovec_, Dec 11 2015
%p evalf(Sum(-Zeta(1, k), k = 2..infinity), 120); # _Vaclav Kotesovec_, Jun 18 2021
%t Sum[ -Zeta'[1 + k], {k, 1, Infinity}] (* _Vladimir Reshetnikov_, Dec 28 2008 *)
%t Integrate[EulerGamma/x + PolyGamma[0, 1+x]/x, {x, 0, 1}] // N[#, 105]& // RealDigits[#][[1]]& (* or *) Integrate[x*Log[x]/((1-x)*Log[1-x]), {x, 0, 1}] // N[#, 105]& // RealDigits[#][[1]]& (* _Jean-François Alcover_, Feb 04 2013 *)
%t $MaxExtraPrecision = 200; NIntegrate[HarmonicNumber[t]/t, {t, 0, 1}, WorkingPrecision -> 105] (* _Yuriy Sibirmovsky_, Sep 04 2016 *)
%t digits = 120; RealDigits[NSum[(-1)^(n + 1)*Zeta[n + 1]/n, {n,1,Infinity}, NSumTerms -> 20*digits, WorkingPrecision -> 10*digits, Method -> "AlternatingSigns"], 10, digits][[1]] (* _G. C. Greubel_, Nov 15 2018 *)
%o (PARI) sumalt(s=1, (-1)^(s+1)/s*zeta(s+1) )
%o (PARI) suminf(k=2, -zeta'(k)) \\ _Vaclav Kotesovec_, Jun 17 2021
%o (Magma) SetDefaultRealField(RealField(100)); L:=RiemannZeta(); (&+[(-1)^(n+1)*Evaluate(L,n+1)/n: n in [1..10^3]]); // _G. C. Greubel_, Nov 15 2018
%o (SageMath) numerical_approx(sum((-1)^(k+1)*zeta(k+1)/k for k in [1..1000]), digits=100) # _G. C. Greubel_, Nov 15 2018
%Y Cf. A000005, A002210, A027423, A075887, A131385, A244109, A001620, A085361, A345682.
%K nonn,cons
%O 1,2
%A _R. J. Mathar_, Sep 14 2007
%E Extended to 105 digits by _Jean-François Alcover_, Feb 04 2013
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