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%I #24 Oct 23 2023 17:38:56
%S 1,4,11,29,68,160,364,820,1813,3981,8674,18782,40387,86443,184232,
%T 391188,827787,1746443,3674573,7712561,16151933,33757505,70422235,
%U 146659055,304947023,633152157,1312820598,2718674046,5623413203,11618957217,23982175093,49452872529
%N a(n) = Sum_{i=2^(n-1)..2^n-1} sigma_0(i), sigma_0(i) number of divisors of n, n positive integer.
%C 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).
%H Chai Wah Wu, <a href="/A153876/b153876.txt">Table of n, a(n) for n = 1..76</a>
%F a(n) = A085831(n) - A085831(n-1)-1. - _R. J. Mathar_, Jan 05 2009
%F a(n) = Sum_{k>=1} k * A346730(n,k). - _Alois P. Heinz_, Aug 01 2021
%o (PARI) a(n) = sum(i=2^(n-1), 2^n-1, numdiv(i)); \\ _Michel Marcus_, Oct 10 2021
%o (Python)
%o from math import isqrt
%o 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
%Y Cf. A000005, A006218, A095256, A140480, A346730.
%K nonn,easy
%O 1,2
%A _Ctibor O. Zizka_, Jan 03 2009
%E a(14)-a(32) from _Alois P. Heinz_, Aug 01 2021