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A065442 Decimal expansion of Erdős-Borwein constant Sum_{k>=1} 1/(2^k - 1). 55
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 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
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.
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
Equals A066766 - A065443. - Amiram Eldar, Oct 16 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
See A038631 for continued fraction.
Sequence in context: A180318 A004016 A093577 * A368501 A198752 A141462
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 June 13 09:03 EDT 2024. Contains 373383 sequences. (Running on oeis4.)