A244824 Sum of all divisors of all positive integers <= 2^n.

1, 4, 15, 56, 220, 857, 3403, 13535, 54077, 215900, 862954, 3450545, 13802279, 55201838, 220792018, 883134861, 3532518195, 14129951284, 56519699688, 226078355122, 904312961284, 3617249936000, 14468996179294, 57875977567596, 231503907383054, 926015589350438

Offset

0,2

Comments

Has a symmetric representation, the same as A024916.

Links

Chai Wah Wu, Table of n, a(n) for n = 0..75

Formula

a(n) = A024916(A000079(n)).

a(n) ~ 2^(2*n-2) * Pi^2/3. - Vaclav Kotesovec, Oct 23 2023

Example

For n = 2 the sum of all divisors of all positive integers <= 4 is [1] + [1+2] + [1+3] + [1+2+4] = 1 + 3 + 4 + 7 = 15, so a(2) = 15.

Prog

(PARI) a(n) = sum(k=1, 2^n, k*floor(2^n/k) ) \\ Jens Kruse Andersen, Jul 26 2014
(Python)
from math import isqrt
def A244824(n): return -(s:=isqrt(m:=1<<n))**2*(s+1) + sum((q:=m//k)*((k<<1)+q+1) for k in range(1, s+1))>>1 # Chai Wah Wu, Oct 23 2023

Crossrefs

Cf. A000079, A000203, A024916, A196020, A235796, A236104, A237270, A237593.

Sequence in context: A087438 A131497 A174958 * A316592 A077823 A047108

Adjacent sequences: A244821 A244822 A244823 * A244825 A244826 A244827

Keyword

nonn

Author

Omar E. Pol, Jul 06 2014

Extensions

More terms from Jens Kruse Andersen, Jul 26 2014

Status

approved