A066767 a(n) = Sum_{k=1..n} sigma(k)*2^(n-k) where sigma(k) = A000203(k) is the sum of divisors of k.

1, 5, 14, 35, 76, 164, 336, 687, 1387, 2792, 5596, 11220, 22454, 44932, 89888, 179807, 359632, 719303, 1438626, 2877294, 5754620, 11509276, 23018576, 46037212, 92074455, 184148952, 368297944, 736595944, 1473191918, 2946383908

Offset

1,2

Comments

a(n) is the numerator of the unreduced fraction of the n-th partial sum of Sum_{k>=1} sigma(k)/2^k where the denominator of that unreduced fraction is 2^n. The partial sums converge to A066766 = 2.744033...

References

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

Links

Table of n, a(n) for n=1..30.

Steven R. Finch, Digital Search Tree Constants [Broken link]

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

Example

a(1) = 2*(1/2);
a(2) = 4*(1/2 + (1+2)/4) since sigma(1) = 1 and sigma(2) = 1 + 2 = 3;
a(3) = 8*(1/2 + (1+2)/4 + (1+3)/8);
a(4) = 16*(1/2 + (1+2)/4 + (1+3)/8 + (1+2+4)/16).

Prog

(PARI) smv(v)= s=0; for(i=1, matsize(v)[2], s=s+v[i]); s
a(n)= sm=0; for(j=1, n, sm=sm+smv(divisors(j)/2^j)); sm*2^n
(PARI) a(n) = 2^n*(sum(k=1, n, sigma(k)/2^k)); \\ Michel Marcus, Apr 25 2022

Crossrefs

Cf. A000203, A066766.

Sequence in context: A076858 A001215 A335651 * A227200 A027974 A027983

Adjacent sequences: A066764 A066765 A066766 * A066768 A066769 A066770

Keyword

nonn

Author

Randall L Rathbun, Jan 16 2002

Status

approved