A131688 Decimal expansion of the constant Sum_{k>=1} log(k + 1) / (k * (k + 1)).

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, 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, 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

Offset: 1

Author

R. J. Mathar, Sep 14 2007

Keywords

• nonn
• cons

Comments

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

Closely related to A085361 (the exponent in the definition of A085291). - Yuriy Sibirmovsky, Sep 04 2016

Examples

			1.257746886944369630009899830495881528511540890508884868977540833522...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, page 62. [Jean-François Alcover, Mar 21 2013]

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Khristo N. Boyadzhiev, A special constant and series with zeta values and harmonic numbers, arXiv:1903.11141 [math.NT], 2019.
• Mark W. Coffey, Series of zeta values, the Stieltjes constants and a sum S_gamma(n), arXiv:math-ph/0706.0345, 2007-2009, eq (38a).
• Paul Erdős, S. W. Graham, Aleksandar Ivic and Carl Pomerance, On the number of divisors of n!, Analytic Number Theory, Volume 138, Progress in Mathematics pp 337-355.
• Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 538.
• Sofia Kalpazidou, Khintchine's constant for Lüroth representation, Journal of Number Theory, Volume 29, Issue 2, June 1988, Pages 196-205.

Crossrefs

Cf. A000005, A002210, A027423, A075887, A131385, A244109, A001620, A085361, A345682.

Programs

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

Maple

evalf(sum((-1)^(n+1)*Zeta(n+1)/n, n=1..infinity), 120); # Vaclav Kotesovec, Dec 11 2015
evalf(Sum(-Zeta(1, k), k = 2..infinity), 120); # Vaclav Kotesovec, Jun 18 2021

Mathematica

Sum[ -Zeta'[1 + k], {k, 1, Infinity}] (* Vladimir Reshetnikov, Dec 28 2008 *)
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 *)
$MaxExtraPrecision = 200; NIntegrate[HarmonicNumber[t]/t, {t, 0, 1}, WorkingPrecision -> 105] (* Yuriy Sibirmovsky, Sep 04 2016 *)
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 *)

PARI

sumalt(s=1, (-1)^(s+1)/s*zeta(s+1) )

PARI

suminf(k=2, -zeta'(k)) \\ Vaclav Kotesovec, Jun 17 2021

SageMath

numerical_approx(sum((-1)^(k+1)*zeta(k+1)/k for k in [1..1000]), digits=100) # G. C. Greubel, Nov 15 2018

Formula

Equals Sum_{s>=1} (-1)^(s+1)*zeta(s+1)/s.

Equals Sum_{k>=1} -zeta'(1 + k), where Zeta' is the derivative of the Riemann zeta function. - Vladimir Reshetnikov, Dec 28 2008

Equals Sum_{s>=1} log(1+1/s)/s. - Jean-François Alcover, Mar 26 2013

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

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

Extensions

Extended to 105 digits by Jean-François Alcover, Feb 04 2013