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A085361 Decimal expansion of the number c = Sum_{n>=1} (zeta(n+1)-1)/n). 10
7, 8, 8, 5, 3, 0, 5, 6, 5, 9, 1, 1, 5, 0, 8, 9, 6, 1, 0, 6, 0, 2, 7, 6, 3, 2, 3, 4, 5, 4, 5, 5, 4, 6, 6, 6, 4, 7, 2, 7, 4, 9, 6, 6, 8, 2, 2, 3, 2, 8, 1, 6, 4, 9, 7, 5, 5, 1, 5, 6, 4, 0, 2, 3, 0, 1, 7, 8, 0, 6, 4, 3, 5, 6, 3, 3, 0, 1, 6, 2, 2, 8, 7, 4, 7, 1, 5, 9, 2, 1, 3, 3, 2, 2, 4, 3, 1, 9, 6, 7, 5, 6 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The Alladi-Grinstead constant (A085291) is exp(c-1).

REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8.1 Alternative representations [of real numbers], p. 62.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Eric Weisstein's World of Mathematics, Alladi-Grinstead Constant

Eric Weisstein's World of Mathematics, Convergence Improvement

FORMULA

c = Sum_{n>=2} log(n/(n-1))/n = Sum_{n>=1, k>=2} 1/(n*k^(n+1)). [From Mathworld links]

EXAMPLE

0.78853056591150896106027632345455466647274966822328164975515640230178...

MAPLE

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

MATHEMATICA

Sum[(-1+Zeta[1+n])/n, {n, Infinity}]

NSum[Log[k]/(k*(k+1)), {k, 1, Infinity}, WorkingPrecision -> 120, NSumTerms ->5000, Method -> {NIntegrate, MaxRecursion -> 100}] (* Vaclav Kotesovec, Dec 11 2015 *)

PROG

(PARI) suminf(n=1, (zeta(n+1)-1-2^(-n-1))/n)+log(2)/2 \\ Charles R Greathouse IV, Feb 20 2012

(Sage)

import mpmath

mpmath.mp.pretty=True; mpmath.mp.dps=108 #precision

mpmath.nsum(lambda n: (-1+mpmath.zeta(1+n))/n, [1, mpmath.inf]) # Peter Luschny, Jul 14 2012

(Sage) numerical_approx(sum((zeta(k+1)-1)/k for k in [1..1000]), digits=120) # G. C. Greubel, Nov 15 2018

(MAGMA) SetDefaultRealField(RealField(120)); L:=RiemannZeta(); (&+[(Evaluate(L, n+1)-1)/n: n in [1..1000]]); // G. C. Greubel, Nov 15 2018

CROSSREFS

Cf. A002210, A085291, A244109.

Sequence in context: A065470 A338815 A197810 * A256781 A248224 A092290

Adjacent sequences:  A085358 A085359 A085360 * A085362 A085363 A085364

KEYWORD

nonn,cons

AUTHOR

Eric W. Weisstein, Jun 25 2003

STATUS

approved

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Last modified December 1 06:09 EST 2020. Contains 338833 sequences. (Running on oeis4.)