A066766 -id:A066766 - cp's OEIS frontend

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

N. J. A. Sloane, Nov 18 2001

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

			1.60669515241529176378330152319092458048057967150575643577807955369...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.
Paul Halmos, "Problems for Mathematicians, Young and Old", Dolciani Mathematical Expositions, 1991, p. 258.

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.

See A038631 for continued fraction.

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

# 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

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 *)

a(n)= s=0; for(x=1,n,s=s+1.0/(2^x-1)); s

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

k=1.; suminf(n=1, k>>=1; k^n*(1+k)/(1-k)) \\ Charles R Greathouse IV, Jun 03 2015

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

More terms from Randall L Rathbun, Jan 16 2002

A065443 Decimal expansion of Sum_{k=1..inf} 1/(2^k-1)^2.

Original entry on oeis.org

1, 1, 3, 7, 3, 3, 8, 7, 3, 6, 3, 4, 4, 1, 9, 6, 5, 9, 6, 6, 9, 6, 9, 1, 3, 3, 6, 8, 3, 0, 1, 3, 4, 7, 5, 8, 3, 8, 3, 0, 8, 4, 9, 3, 0, 9, 8, 1, 3, 8, 8, 2, 8, 8, 2, 0, 7, 0, 4, 4, 9, 3, 3, 1, 0, 4, 6, 4, 9, 3, 8, 6, 2, 5, 2, 0, 4, 0, 8, 9, 9, 8, 0, 0, 0, 5, 4, 0, 5, 0, 9, 0, 4, 2, 3, 5, 1, 3, 1, 1, 8, 4, 0, 3, 6

Offset: 1

N. J. A. Sloane, Nov 18 2001

			1.1373387363441965966969133683013475838308493098...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.

Harry J. Smith, Table of n, a(n) for n=1..2000
Steven R. Finch, Digital Search Tree Constants. [Broken link]
Steven R. Finch, Digital Search Tree Constants. [From the Wayback machine]
Eric Weisstein's World of Mathematics, Tree Searching.

Cf. A060867, A065442, A065608, A066766, A067616.

RealDigits[NSum[1/(2^k - 1)^2, {k, 1, Infinity}, PrecisionGoal -> 40, AccuracyGoal -> 40, WorkingPrecision -> 500, NSumTerms -> 50, NSumExtraTerms -> 50]][[1]] (* Peter Bertok (peter(AT)bertok.com), Dec 04 2001 *)
RealDigits[(Log[2] QPolyGamma[0, 1, 1/2] + QPolyGamma[1, 1, 1/2])/Log[2]^2 - 1, 10, 20][[1]] (* Eric W. Weisstein, Jun 02 2025 *)

{ default(realprecision, 2080); x=suminf(k=1, 1/(2^k - 1)^2); for (n=1, 2000, d=floor(x); x=(x-d)*10; write("b065443.txt", n, " ", d)) } \\ Harry J. Smith, Oct 19 2009

Equals Sum_{n>=1} 1/A060867(n).
From Amiram Eldar, Oct 16 2022: (Start)
Equals Sum_{k>=1} k/(2^(k+1)-1).
Equals A066766 - A065442. (End)
Equals Sum_{n >= 1} q^(n^2)*( (n - 1) + q^n - (n - 1)*q^(2*n) )/(1 - q^n)^2 evaluated at q = 1/2 (see A065608). - Peter Bala, Oct 16 2022

More terms from Peter Bertok (peter(AT)bertok.com), Dec 04 2001

A335764 Decimal expansion of Sum_{k>=1} sigma(k)/(k*2^k) where sigma(k) is the sum of divisors of k (A000203).

Original entry on oeis.org

1, 2, 4, 2, 0, 6, 2, 0, 9, 4, 8, 1, 2, 4, 1, 4, 9, 4, 5, 7, 9, 7, 8, 4, 5, 4, 8, 1, 8, 9, 4, 6, 2, 9, 6, 6, 8, 9, 7, 3, 4, 0, 3, 9, 7, 8, 2, 5, 0, 4, 2, 5, 8, 8, 4, 6, 2, 7, 1, 3, 8, 1, 6, 7, 2, 5, 3, 3, 9, 1, 1, 8, 4, 4, 7, 0, 6, 2, 8, 8, 4, 6, 5, 8, 2, 4, 1

Offset: 1

Amiram Eldar, Jun 21 2020

			1.242062094812414945797845481894629668973403978250425...

Maxie Dion Schmidt, A catalog of interesting and useful Lambert series identities, arXiv:2004.02976 [math.NT], 2020.
Eric Weisstein's World of Mathematics, Lambert Series.
Wikipedia, Lambert series.

Cf. A000203, A017665, A017666, A048651, A065442, A066524, A066766, A256936, A335763.

RealDigits[Sum[1/n/(2^n - 1), {n, 1, 500}], 10, 100][[1]]
RealDigits[-Log[QPochhammer[1/2]], 10, 120][[1]] (* Vaclav Kotesovec, Feb 18 2021 *)

suminf(x = 1, sigma(x)/(x*2^x)) \\ David A. Corneth, Jun 21 2020

Equals Sum_{k>=1} (A017665(k)/A017666(k))/2^k.
Equals Sum_{k>=1} 1/(k*(2^k - 1)) = Sum_{k>=1} 1/A066524(k).
Equals -Sum_{k>=1} log(1-2^(-k)).
Equals -log(A048651). - Amiram Eldar, Feb 19 2022

A335763 Decimal expansion of Sum_{k>=1} sigma_2(k)/2^k where sigma_2(k) is the sum of squares of divisors of k (A001157).

Original entry on oeis.org

7, 0, 9, 9, 2, 8, 5, 1, 7, 8, 8, 9, 0, 9, 0, 7, 1, 1, 4, 0, 3, 3, 1, 2, 5, 0, 2, 2, 1, 6, 4, 7, 5, 3, 6, 6, 3, 1, 5, 7, 6, 0, 8, 8, 3, 3, 2, 1, 1, 8, 9, 5, 9, 7, 8, 8, 3, 9, 2, 3, 7, 7, 4, 2, 8, 8, 9, 1, 2, 8, 8, 9, 1, 1, 2, 2, 6, 4, 5, 8, 7, 1, 7, 3, 5, 5, 4

Offset: 1

Amiram Eldar, Jun 21 2020

			7.099285178890907114033125022164753663157608833211895...

Maxie Dion Schmidt, A catalog of interesting and useful Lambert series identities, arXiv:2004.02976 [math.NT], 2020.
Eric Weisstein's World of Mathematics, Lambert Series.
Wikipedia, Lambert series.

Cf. A001157, A065442, A066766, A227989, A256936, A335764.

evalf(add( (1/2)^(n^2)*( n^2*(8^n) - ((n-1)^2 - 2)*4^n - ((n+1)^2 - 2)*(2^n) + n^2 )/(2^n - 1)^3, n = 1..20 ), 100); # Peter Bala, Jan 22 2021

RealDigits[Sum[n^2/(2^n - 1), {n, 1, 500}], 10, 100][[1]]

Equals Sum_{k>=1} k^2/(2^k - 1).

Faster converging series: Sum_{n >= 1} (1/2)^(n^2)*( n^2*(8^n) - ((n-1)^2 - 2)*4^n - ((n+1)^2 - 2)*(2^n) + n^2 )/(2^n - 1)^3. - Peter Bala, Jan 19 2021

A066772 Continued fraction expansion for Sum_{k>=1} d(k)/2^k where d(k) are divisors of k, 1 <= d <= k.

Original entry on oeis.org

2, 1, 2, 1, 9, 1, 2, 1, 1, 1, 5, 8, 1, 3, 2, 3, 2, 2, 2, 2, 1, 3, 3, 4, 12, 1, 1, 1, 11, 4, 15, 6, 3, 3, 2, 1, 20, 4, 4, 1, 4, 1, 2, 1, 2, 6, 1, 2, 1, 28, 107, 1, 4, 4, 3, 1, 2, 2, 4, 2, 3, 51, 1, 1, 1, 4, 2, 4, 1, 20, 20, 1, 14, 3, 27, 1, 4, 3, 14, 329, 1, 1, 1, 111, 2, 3, 1, 2, 1, 4, 1, 6, 1, 4, 1

Offset: 0

Randall L Rathbun, Jan 16 2002

A(1669) accurate to 500 decimal digits.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.

Steven R. Finch, Digital Search Tree Constants [Broken link]
Steven R. Finch, Digital Search Tree Constants [From the Wayback machine]

Cf. A066766 (decimal expansion).

{smv(v)= s=0; for(i=1,matsize(v)[2],s=s+v[i]); s }
{A066766(n)= sm=0; for(j=1,n,sm=sm+smv(divisors(j)/2^j)); sm*1.0 }

contfrac(suminf(k=1, sigma(k)/2^k)) \\ Michel Marcus, Apr 25 2022

Offset changed by Andrew Howroyd, Jul 04 2024

A066767 a(n) = Sum_{k=1..n} sigma(k)*2^(n-k) where sigma(k) = A000203(k) is the sum of divisors of k.

Original entry on oeis.org

1, 5, 14, 35, 76, 164, 336, 687, 1387, 2792, 5596, 11220, 22454, 44932, 89888, 179807, 359632, 719303, 1438626, 2877294, 5754620, 11509276, 23018576, 46037212, 92074455, 184148952, 368297944, 736595944, 1473191918, 2946383908

Offset: 1

Randall L Rathbun, Jan 16 2002

nonn

a(n) is the numerator of the unreduced fraction of the n-th partial sum of Sum_{k>=1} sigma(k)/2^k where the denominator of that unreduced fraction is 2^n. The partial sums converge to A066766 = 2.744033...

			a(1) = 2*(1/2);
a(2) = 4*(1/2 + (1+2)/4) since sigma(1) = 1 and sigma(2) = 1 + 2 = 3;
a(3) = 8*(1/2 + (1+2)/4 + (1+3)/8);
a(4) = 16*(1/2 + (1+2)/4 + (1+3)/8 + (1+2+4)/16).

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.

Steven R. Finch, Digital Search Tree Constants [Broken link]
Steven R. Finch, Digital Search Tree Constants [From the Wayback machine]

Cf. A000203, A066766.

smv(v)= s=0; for(i=1,matsize(v)[2],s=s+v[i]); s
a(n)= sm=0; for(j=1,n,sm=sm+smv(divisors(j)/2^j)); sm*2^n

a(n) = 2^n*(sum(k=1, n, sigma(k)/2^k)); \\ Michel Marcus, Apr 25 2022

A335765 Decimal expansion of Sum_{k>=1} 1/2^(k-omega(k)) where omega(k) is the number of distinct primes dividing k (A001221).

Original entry on oeis.org

1, 5, 3, 3, 8, 8, 5, 6, 4, 1, 4, 7, 4, 3, 8, 0, 6, 6, 6, 8, 2, 6, 4, 0, 6, 0, 3, 0, 9, 7, 0, 6, 3, 2, 8, 8, 1, 5, 0, 0, 7, 0, 7, 9, 4, 0, 3, 6, 2, 1, 5, 4, 7, 7, 9, 1, 6, 6, 3, 3, 8, 1, 2, 5, 8, 9, 8, 0, 9, 4, 8, 9, 6, 3, 8, 0, 4, 3, 8, 8, 6, 4, 4, 3, 9, 5, 4

Offset: 1

Amiram Eldar, Jun 21 2020

			1.533885641474380666826406030970632881500707940362154...

Maxie Dion Schmidt, A catalog of interesting and useful Lambert series identities, arXiv:2004.02976 [math.NT], 2020.
Eric Weisstein's World of Mathematics, Lambert Series.
Wikipedia, Lambert series.

Cf. A001221, A000225, A005117, A008683, A034444, A065442, A066766, A256936, A335763, A335764.

RealDigits[Sum[1/2^(n - PrimeNu[n]), {n, 1, 500}], 10, 100][[1]]

Equals Sum_{k>=1} A034444(k)/2^k.

Equals Sum_{k>=1} mu(k)^2/(2^k - 1), where mu(k) is the Möbius function (A008683), or, equivalently, Sum_{k>=1} 1/A000225(A005117(k)).

A367325 Decimal expansion of Sum_{k>=1} sigma(k)/(2^k-1), where sigma(k) is the sum of divisors of k (A000203).

Original entry on oeis.org

3, 6, 0, 4, 4, 4, 7, 3, 4, 1, 9, 7, 1, 9, 4, 6, 7, 4, 4, 8, 9, 3, 6, 4, 8, 4, 7, 3, 6, 2, 3, 5, 8, 8, 3, 5, 6, 0, 0, 4, 9, 5, 4, 8, 7, 0, 6, 4, 9, 9, 8, 7, 5, 0, 1, 3, 8, 3, 6, 2, 6, 3, 1, 5, 0, 9, 6, 6, 2, 9, 5, 0, 1, 7, 0, 7, 1, 3, 4, 9, 6, 6, 9, 1, 7, 8, 2, 2, 8, 9, 9, 1, 6, 9, 9, 2, 7, 5, 4, 4, 2, 7, 0, 3, 8

Offset: 1

Amiram Eldar, Nov 14 2023

			3.60444734197194674489364847362358835600495487064998...

Maxie Dion Schmidt, A catalog of interesting and useful Lambert series identities, arXiv:2004.02976 [math.NT], 2020. See eq. (6.1c), p. 13.
Michael I. Shamos, Shamos's catalog of the real numbers, 2011. See p. 526.

Cf. A000005, A000203.

Similar constants: A065442, A066766, A116217, A335763, A335764.

with(numtheory): evalf(sum(sigma(k)/(2^k-1), k = 1..infinity), 120)

RealDigits[Sum[DivisorSigma[1,n]/(2^n-1), {n, 1, 500}], 10, 120][[1]]

PARI
```
suminf(k = 1, sigma(k)/(2^k-1))
```

suminf(k = 1, numdiv(k)/((2^k-1)*(1-1/2^k)))

Equals Sum_{k>=1} d(k)/((2^k-1)*(1-1/2^k)), where d(k) is the number of divisors of k (A000005).

cp's OEIS Frontend

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

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A065443 Decimal expansion of Sum_{k=1..inf} 1/(2^k-1)^2.

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A335764 Decimal expansion of Sum_{k>=1} sigma(k)/(k*2^k) where sigma(k) is the sum of divisors of k (A000203).

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A335763 Decimal expansion of Sum_{k>=1} sigma_2(k)/2^k where sigma_2(k) is the sum of squares of divisors of k (A001157).

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A066772 Continued fraction expansion for Sum_{k>=1} d(k)/2^k where d(k) are divisors of k, 1 <= d <= k.

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A066767 a(n) = Sum_{k=1..n} sigma(k)*2^(n-k) where sigma(k) = A000203(k) is the sum of divisors of k.

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A335765 Decimal expansion of Sum_{k>=1} 1/2^(k-omega(k)) where omega(k) is the number of distinct primes dividing k (A001221).

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