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A131688 Decimal expansion of the constant Sum_{k>=1} log(k+1)/k/(k+1). 7
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Equals Sum[ -Zeta'[1 + k], {k, 1, Infinity}], where Zeta' is the derivative of Riemann Zeta function. - Vladimir Reshetnikov, Dec 28 2008

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

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

M. 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 Erdos, 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.

Sofia Kalpazidou, Khintchine's constant for Lüroth representation, Journal of Number Theory, Volume 29, Issue 2, June 1988, Pages 196-205.

FORMULA

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

Also 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

EXAMPLE

1.257746886944369630009899830495881528511540890508884868977540833522...

MAPLE

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

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 *)

PROG

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

(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

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

CROSSREFS

Cf. A002210, A075887, A131385, A244109, A001620, A085361.

Sequence in context: A325437 A141430 A021392 * A226213 A199590 A096624

Adjacent sequences:  A131685 A131686 A131687 * A131689 A131690 A131691

KEYWORD

cons,nonn

AUTHOR

R. J. Mathar, Sep 14 2007

EXTENSIONS

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

STATUS

approved

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Last modified August 22 08:18 EDT 2019. Contains 326172 sequences. (Running on oeis4.)