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

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

Author

Amiram Eldar, Jun 21 2020

Keywords

• nonn
• cons

Examples

			1.533885641474380666826406030970632881500707940362154...
		

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

Programs

Mathematica

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

Formula

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)).