OFFSET
1,2
COMMENTS
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
This constant is irrational (Erdős, 1948; Borwein, 1992). - Amiram Eldar, Aug 01 2020
REFERENCES
Steven 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
David H. Bailey and Richard E. Crandall, Random generators and normal numbers, Experimental Mathematics, Vol. 11, No. 4 (2002), pp. 527-546.
Robert Baillie, Summing The Curious Series Of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015.
Peter Borwein, On the Irrationality of Certain Series, Math. Proc. Cambridge Philos. Soc., Vol. 112, No. 1 (1992), pp. 141-146, alternative link.
Richard Crandall, The googol-th bit of the Erdős-Borwein constant, Integers, 12 (2012), A23.
Paul Erdős, On Arithmetical Properties of Lambert Series, J. Indian Math. Soc., Vol. 12 (1948), 63-66.
Steven R. Finch, Digital Search Tree Constants [Broken link]
Steven R. Finch, Digital Search Tree Constants [From the Wayback machine]
Nobushige Kurokawa and Yuichiro Taguchi, A p-analogue of Euler’s constant and congruence zeta functions, Proc. Japan Acad. Ser. A Math. Sci., Volume 94, Number 2 (2018), 13-16.
Mathematics Stack Exchange, Find Sum_{k = 1..oo} 1/(2^(k+1) - 1).
Yohei Tachiya, Irrationality of Certain Lambert Series, Tokyo J. Math. 27 (1) 75 - 85, June 2004.
László Tóth, Alternating sums concerning multiplicative arithmetic functions, arXiv preprint arXiv:1608.00795 [math.NT], 2016.
Eric Weisstein's Mathworld, Erdos-Borwein Constant, Tree Searching, Double Series, Irrational Number
Hengjie Yang and Richard D. Wesel, Systematic Transmission With Fountain Parity Checks for Erasure Channels With Stop Feedback, arXiv:2307.14507 [cs.IT], 2023.
Rimer Zurita, Generalized Alternating Sums of Multiplicative Arithmetic Functions, J. Int. Seq., Vol. 23 (2020), Article 20.10.4.
FORMULA
Note: Sum_{k>=1} d(k)/2^k = Sum_{k>=1} 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
Equals (1/2) * A211705. - Amiram Eldar, Aug 01 2020
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
EXAMPLE
1.60669515241529176378330152319092458048057967150575643577807955369...
MAPLE
# Uses Lambert series, cf. formula by Arndt:
evalf( add( (1/2)^(n^2)*(1 + 2/(2^n - 1)), n = 1..20 ), 105);
# Peter Bala, Jan 22 2021
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) a(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) k=1.; suminf(n=1, k>>=1; k^n*(1+k)/(1-k)) \\ Charles R Greathouse IV, Jun 03 2015
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
N. J. A. Sloane, Nov 18 2001
EXTENSIONS
More terms from Randall L Rathbun, Jan 16 2002
STATUS
approved