A153876 a(n) = Sum_{i=2^(n-1)..2^n-1} sigma_0(i), sigma_0(i) number of divisors of n, n positive integer.

1, 4, 11, 29, 68, 160, 364, 820, 1813, 3981, 8674, 18782, 40387, 86443, 184232, 391188, 827787, 1746443, 3674573, 7712561, 16151933, 33757505, 70422235, 146659055, 304947023, 633152157, 1312820598, 2718674046, 5623413203, 11618957217, 23982175093, 49452872529

Offset: 1

Author

Ctibor O. Zizka, Jan 03 2009

Keywords

• nonn
• easy

Comments

This sequence tells how many binary numbers with n digits are there in the multiplication matrix [1,...,2^n -1]x[1,...,2^n -1]. In general, counting how many base-B numbers of length n are there in the multiplication matrix [1,...,B^n -1]x[1,...,B^n -1] gives a(n)= sum_{i=B^(n-1),(B^n)-1} sigma_0(i). Besides this motivation it is interesting to see the behavior of partial sums of sigma_0(i) on growing intervals : a(n)= sum_{i=f(n-1),f(n)} sigma_0(i).

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..76

Crossrefs

Cf. A000005, A006218, A095256, A140480, A346730.

Programs

PARI

a(n) = sum(i=2^(n-1), 2^n-1, numdiv(i)); \\ Michel Marcus, Oct 10 2021

Python

from math import isqrt
def A153876(n): return ((t:=isqrt(b:=(1<Chai Wah Wu, Oct 23 2023

Formula

a(n) = A085831(n) - A085831(n-1)-1. - R. J. Mathar, Jan 05 2009

a(n) = Sum_{k>=1} k * A346730(n,k). - Alois P. Heinz, Aug 01 2021

Extensions

a(14)-a(32) from Alois P. Heinz, Aug 01 2021