A065442 Decimal expansion of Erdős-Borwein constant Sum_{k>=1} 1/(2^k - 1).

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

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

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.

Cf. A000005, A000203 (see formulas), A065443, A066766, A179951, A211705, A211706, A323482.

Sequence in context: A180318 A004016 A093577 * A368501 A198752 A141462

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