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

Author

Randall L Rathbun, Jan 16 2002

Keywords

• nonn
• cons

Examples

			2.74403388875948836048021489149227216431142898131963931784...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.

Links

• Steven R. Finch, Digital Search Tree Constants [Broken link]
• Steven R. Finch, Digital Search Tree Constants [From the Wayback machine]

Crossrefs

Cf. A000203, A065442, A065443, A066772, A335763.

Programs

Maple

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

Mathematica

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

PARI

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

PARI

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

Formula

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)

Extensions

Name corrected by Paul D. Hanna, Apr 26 2018