OFFSET
1,2
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
FORMULA
a(n) = Sum_{k>=1} k * A346730(n,k). - Alois P. Heinz, Aug 01 2021
PROG
(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<<n-1)-1))+(s:=isqrt(a:=(1<<n)-1)))*(t-s)+(sum(a//k for k in range(1, s+1))-sum(b//k for k in range(1, t+1))<<1) # Chai Wah Wu, Oct 23 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ctibor O. Zizka, Jan 03 2009
EXTENSIONS
a(14)-a(32) from Alois P. Heinz, Aug 01 2021
STATUS
approved