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%I #117 Feb 08 2024 01:40:52
%S 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,
%T 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,
%U 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
%N Decimal expansion of Erdős-Borwein constant Sum_{k>=1} 1/(2^k - 1).
%C Also the decimal expansion of the (finite) value of Sum_{ k >= 1, k has no digit equal to 0 in base 2 } 1/k. - _Robert G. Wilson v_, Aug 03 2010
%C This constant is irrational (Erdős, 1948; Borwein, 1992). - _Amiram Eldar_, Aug 01 2020
%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.
%D Paul Halmos, "Problems for Mathematicians, Young and Old", Dolciani Mathematical Expositions, 1991, p. 258.
%H Harry J. Smith, <a href="/A065442/b065442.txt">Table of n, a(n) for n = 1..2000</a>
%H David H. Bailey and Richard E. Crandall, <a href="https://projecteuclid.org/euclid.em/1057864662">Random generators and normal numbers</a>, Experimental Mathematics, Vol. 11, No. 4 (2002), pp. 527-546.
%H Robert Baillie, <a href="http://arxiv.org/abs/0806.4410">Summing The Curious Series Of Kempner and Irwin</a>, arXiv:0806.4410 [math.CA], 2008-2015.
%H Peter Borwein, <a href="https://doi.org/10.1017/S030500410007081X">On the Irrationality of Certain Series</a>, Math. Proc. Cambridge Philos. Soc., Vol. 112, No. 1 (1992), pp. 141-146, <a href="http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P59.pdf">alternative link</a>.
%H Richard Crandall, <a href="http://www.emis.de/journals/INTEGERS/papers/m23/m23.Abstract.html">The googol-th bit of the Erdős-Borwein constant</a>, Integers, 12 (2012), A23.
%H Paul Erdős, <a href="https://users.renyi.hu/~p_erdos/1948-04.pdf">On Arithmetical Properties of Lambert Series</a>, J. Indian Math. Soc., Vol. 12 (1948), 63-66.
%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/dig/dig.html">Digital Search Tree Constants</a> [Broken link]
%H Steven R. Finch, <a href="http://web.archive.org/web/20010413214937/http://www.mathsoft.com:80/asolve/constant/dig/dig.html">Digital Search Tree Constants</a> [From the Wayback machine]
%H Nobushige Kurokawa and Yuichiro Taguchi, <a href="http://dx.doi.org/10.3792/pjaa.94.13">A p-analogue of Euler’s constant and congruence zeta functions</a>, Proc. Japan Acad. Ser. A Math. Sci., Volume 94, Number 2 (2018), 13-16.
%H Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/1978310/find-sum-k-1-infty-frac12k1-1">Find Sum_{k = 1..oo} 1/(2^(k+1) - 1)</a>.
%H Yohei Tachiya, <a href="https://doi.org/10.3836/tjm/1244208475">Irrationality of Certain Lambert Series</a>, Tokyo J. Math. 27 (1) 75 - 85, June 2004.
%H László Tóth, <a href="https://arxiv.org/abs/1608.00795">Alternating sums concerning multiplicative arithmetic functions</a>, arXiv preprint arXiv:1608.00795 [math.NT], 2016.
%H Eric Weisstein's Mathworld, <a href="http://mathworld.wolfram.com/Erdos-BorweinConstant.html">Erdos-Borwein Constant</a>, <a href="http://mathworld.wolfram.com/TreeSearching.html">Tree Searching</a>, <a href="http://mathworld.wolfram.com/DoubleSeries.html">Double Series</a>, <a href="http://mathworld.wolfram.com/IrrationalNumber.html">Irrational Number</a>
%H Hengjie Yang and Richard D. Wesel, <a href="https://arxiv.org/abs/2307.14507">Systematic Transmission With Fountain Parity Checks for Erasure Channels With Stop Feedback</a>, arXiv:2307.14507 [cs.IT], 2023.
%H Rimer Zurita, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Zurita/zur3.html">Generalized Alternating Sums of Multiplicative Arithmetic Functions</a>, J. Int. Seq., Vol. 23 (2020), Article 20.10.4.
%F Note: Sum_{k>=1} d(k)/2^k = Sum_{k>=1} 1/(2^k - 1).
%F 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
%F Equals (1/2) * A211705. - _Amiram Eldar_, Aug 01 2020
%F Equals 1/4 + Sum_{k >= 2} (1 + 8^k)/((2^k - 1)*2^(k^2+k)). See Mathematics Stack Exchange link. - _Peter Bala_, Jan 28 2022
%F Equals A066766 - A065443. - _Amiram Eldar_, Oct 16 2022
%e 1.60669515241529176378330152319092458048057967150575643577807955369...
%p # Uses Lambert series, cf. formula by Arndt:
%p evalf( add( (1/2)^(n^2)*(1 + 2/(2^n - 1)), n = 1..20 ), 105);
%p # _Peter Bala_, Jan 22 2021
%t RealDigits[ Sum[1/(2^k - 1), {k, 350}], 10, 111][[1]] (* _Robert G. Wilson v_, Nov 05 2006 *)
%t (* first install irwinSums.m, see reference, then *) First@ RealDigits@ iSum[0, 0, 111, 2] (* _Robert G. Wilson v_, Aug 03 2010 *)
%t RealDigits[(Log[2] - 2 QPolyGamma[0, 1, 2])/Log[4], 10, 100][[1]] (* _Fred Daniel Kline_, May 23 2011 *)
%t 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 *)
%o (PARI) a(n)= s=0; for(x=1,n,s=s+1.0/(2^x-1)); s
%o (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
%o (PARI) k=1.; suminf(n=1, k>>=1; k^n*(1+k)/(1-k)) \\ _Charles R Greathouse IV_, Jun 03 2015
%Y See A038631 for continued fraction.
%Y Cf. A000005, A000203 (see formulas), A065443, A066766, A179951, A211705, A211706, A323482.
%K nonn,cons
%O 1,2
%A _N. J. A. Sloane_, Nov 18 2001
%E More terms from _Randall L Rathbun_, Jan 16 2002
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