A335763 -id:A335763 - cp's OEIS frontend

A066766 Decimal expansion of Sum_{k>=1} sigma(k)/2^k where sigma(k) is the sum of divisors of k, 1 <= d <= k.

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

Offset: 1

Randall L Rathbun, Jan 16 2002

			2.74403388875948836048021489149227216431142898131963931784...

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, A065442, A065443, A066772, A335763.

evalf( add( (1/2)^(n^2) * (n*(4^n - 1) + 2^n)/(2^n - 1)^2, n = 1..20), 100); # Peter Bala, Jan 19 2021

RealDigits[Sum[n/(2^n - 1), {n, 1, 500}], 10, 100][[1]] (* Amiram Eldar, Jun 22 2020 *)

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

suminf(k=1, sigma(k)/2^k) \\ Michel Marcus, Apr 27 2018

Equals Sum_{k>=1} k/(2^k - 1). - Amiram Eldar, Jun 22 2020
Faster converging series: Sum_{n >= 1} (1/2)^(n^2) * (n*(4^n - 1) + 2^n)/(2^n - 1)^2. - Peter Bala, Jan 19 2021
From Amiram Eldar, Oct 16 2022: (Start)
Equals Sum_{k>=1} 2^k/(2^k - 1)^2.
Equals A065442 + A065443. (End)

Name corrected by Paul D. Hanna, Apr 26 2018

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

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

A066766 Decimal expansion of Sum_{k>=1} sigma(k)/2^k where sigma(k) is the sum of divisors of k, 1 <= d <= k.

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

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Programs

Maple

Mathematica

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