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A065442
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Decimal expansion of Erdos-Borwein constant Sum_{k=1..inf} 1/(2^k-1).
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10
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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
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OFFSET
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1,2
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COMMENTS
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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. [From Robert G. Wilson v, Aug 03 2010]
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.
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LINKS
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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
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FORMULA
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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]
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EXAMPLE
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1.60669515241529176378330152319092458048057967150575643577807955369...
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MATHEMATICA
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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 *)
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PROG
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(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)) } [From Harry J. Smith, Oct 19 2009]
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CROSSREFS
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See A038631 for continued fraction.
Sequence in context: A004016 A180318 A093577 * A198752 A141462 A055955
Adjacent sequences: A065439 A065440 A065441 * A065443 A065444 A065445
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KEYWORD
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nonn,cons
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AUTHOR
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N. J. A. Sloane, Nov 18 2001
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EXTENSIONS
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More terms from Randall L. Rathbun, Jan 16 2002
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STATUS
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approved
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