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).

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

1,1

Links

Table of n, a(n) for n=1..87.

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} 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

Example

7.099285178890907114033125022164753663157608833211895...

Maple

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

Mathematica

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

Crossrefs

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

Sequence in context: A073115 A176444 A197025 * A096408 A005481 A122699

Adjacent sequences: A335760 A335761 A335762 * A335764 A335765 A335766

Keyword

nonn,cons

Author

Amiram Eldar, Jun 21 2020

Status

approved