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

Author

Randall L Rathbun, Jan 16 2002

Keywords

• nonn

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...

Examples

			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).
		

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, A066766.

Programs

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