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A065442 Decimal expansion of Erdős-Borwein constant Sum_{k=1..inf} 1/(2^k-1). 26
1, 6, 0, 6, 6, 9, 5, 1, 5, 2, 4, 1, 5, 2, 9, 1, 7, 6, 3, 7, 8, 3, 3, 0, 1, 5, 2, 3, 1, 9, 0, 9, 2, 4, 5, 8, 0, 4, 8, 0, 5, 7, 9, 6, 7, 1, 5, 0, 5, 7, 5, 6, 4, 3, 5, 7, 7, 8, 0, 7, 9, 5, 5, 3, 6, 9, 1, 4, 1, 8, 4, 2, 0, 7, 4, 3, 4, 8, 6, 6, 9, 0, 5, 6, 5, 7, 1, 1, 8, 0, 1, 6, 7, 0, 1, 5, 5, 5, 7, 5, 8, 9, 7, 0, 4 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Also the decimal expansion of the (finite) value of the sum_{ k >= 1, k has no digit equal to 0 in base 2 } 1/k. - Robert G. Wilson v, Aug 03 2010

REFERENCES

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.

Paul Halmos, "Problems for Mathematicians, Young and Old", Dolciani Mathematical Expositions, 1991, p. 258.

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..2000

Robert Baillie, Summing The Curious Series Of Kempner and Irwin,  arXiv:0806.4410v2 [math.CA], (2008)

Richard Crandall, The googol-th bit of the Erdos-Borwein constant, Integers, 12 (2012), A23.

S. R. Finch, Digital Search Tree Constants,

Eric Weisstein's Mathworld, Erdos-Borwein Constant, Tree Searching, Double Series, Irrational Number

FORMULA

Note Sum_{k=1..inf} d(k)/2^k = Sum_{k=1..inf} 1/(2^k-1).

Fast computation via Lambert series: 1.60669515... = sum(n>=1, x^(n^2)*(1+x^n)/(1-x^n) ) where x=1/2. - Joerg Arndt, May 24 2011

Sum_{k=0..inf} 1/sigma(2^k) = 1.60669515... - Paolo P. Lava, Feb 10 2014

EXAMPLE

1.60669515241529176378330152319092458048057967150575643577807955369...

MATHEMATICA

RealDigits[ Sum[1/(2^k - 1), {k, 350}], 10, 111][[1]] (* Robert G. Wilson v, Nov 05 2006 *)

(* first install irwinSums.m, see reference, then *) First@ RealDigits@ iSum[0, 0, 111, 2] (* Robert G. Wilson v, Aug 03 2010 *)

RealDigits[(Log[2] - 2 QPolyGamma[0, 1, 2])/Log[4], 10, 100][[1]] (* Fred Daniel Kline, May 23 2011 *)

x = 1/2; RealDigits[ Sum[ DivisorSigma[0, k] x^k, {k, 1000}], 10, 105][[1]] (* Robert G. Wilson v, Oct 12 2014 after an observation and formula of Amarnath Murthy, see A073668 *)

PROG

(PARI) A065442(n)= s=0; for(x=1, n, s=s+1.0/(2^x-1)); s

(PARI) default(realprecision, 2080); x=suminf(k=1, 1/(2^k - 1)); for (n=1, 2000, d=floor(x); x=(x-d)*10; write("b065442.txt", n, " ", d)) \\ Harry J. Smith, Oct 19 2009

(PARI) suminf(n=1, my(k=.5^n); k^n*(1+k)/(1-k)) \\ Charles R Greathouse IV, Jun 03 2015

CROSSREFS

See A038631 for continued fraction.

Sequence in context: A180318 A004016 A093577 * A198752 A141462 A055955

Adjacent sequences:  A065439 A065440 A065441 * A065443 A065444 A065445

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane, Nov 18 2001

EXTENSIONS

More terms from Randall L. Rathbun, Jan 16 2002

STATUS

approved

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Last modified December 6 15:11 EST 2016. Contains 278781 sequences.