login
A335764
Decimal expansion of Sum_{k>=1} sigma(k)/(k*2^k) where sigma(k) is the sum of divisors of k (A000203).
5
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
1,2
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.
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
EXAMPLE
1.242062094812414945797845481894629668973403978250425...
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 *)
PROG
(PARI) suminf(x = 1, sigma(x)/(x*2^x)) \\ David A. Corneth, Jun 21 2020
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Jun 21 2020
STATUS
approved