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

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

Author

Amiram Eldar, Jun 21 2020

Keywords

• nonn
• cons

Examples

			1.242062094812414945797845481894629668973403978250425...
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

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 *)

PARI

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

Formula

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