A095256 -id:A095256 - cp's OEIS frontend

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

Ctibor O. Zizka, Jan 03 2009

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).

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

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

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

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

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

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

A153816 a(n) = Sum_{i=1..(10^n-1)/9} floor(((10^n-1)/9)/i).

Original entry on oeis.org

1, 29, 542, 7967, 105225, 1308095, 15639310, 181976675, 2075608136, 23314508721, 258729364359, 2843136431305, 30989792180446, 335482200606705, 3610664794156597, 38665075822637767, 412235037037411453, 4378193158484415385, 46340359465948601163

Offset: 1

Ctibor O. Zizka, Jan 02 2009

Generalized subsequences of A006218(n) are a(n) = A006218(T*A002275(n)), where T >= 1, a(n) = Sum_{i=1...n} floor(T*(10^n - 1)/9*i). For T=9 we have A095256, for T=1 this sequence. The motivation for such sequences is to count the elements of length n in a multiplication matrix m*m in base (T+1). In base 10 this gives T=9 and the number of elements of the multiplication matrix m*m of the length n=1,2,3,... digits is given by the sequence b(n) = a(n) - a(n-1), n >= 2, a(1)=23.

Chai Wah Wu, Table of n, a(n) for n = 1..26

Cf. A000005, A095256, A006218, A002275.

A153816[n_] := Sum[Floor[((10^n - 1)/9)/i], {i, (10^n - 1)/9}]; Array[A153816, 7] (* JungHwan Min, Feb 05 2017 *)

a(n) = sum(i=1, (10^n-1)/9, ((10^n-1)/9)\i); \\ Michel Marcus, Jun 08 2018

def a(n): t = (10**n-1)//9; return sum(t//i for i in range(1, t+1))
print([a(n) for n in range(1, 10)]) # Michael S. Branicky, Jan 29 2021

from math import isqrt
def A153816(n): return -(s:=isqrt(m:=(10**n-1)//9))**2+(sum(m//k for k in range(1,s+1))<<1) # Chai Wah Wu, Oct 23 2023

a(n) = A006218(A002275(n)).

Formula corrected by Giovanni Resta, Feb 05 2009
a(9)-a(17) from Donovan Johnson, Sep 06 2010
a(18)-a(19) from Hiroaki Yamanouchi, Jul 06 2014

A174425 Total number of divisors of all n-digit numbers.

Original entry on oeis.org

23, 450, 6580, 86590, 1073071, 12803271, 148755315, 1694786187, 19020186047, 210925125565, 2316483913054, 25237165712764, 273094922940644, 2938181887791268, 31454145461543161, 335264720452385137, 3559879862893130917, 37671125212625723995, 397434517963203503069

Offset: 1

Jaroslav Krizek, Nov 28 2010

Partial sums are A095256.

			For n = 1; a(1) = 23 because tau (r) of 1-digit numbers r = 1 to 9: {1, 2, 2, 3, 2, 4, 2, 4, 3}. Sum is 23.

Andrew Howroyd, Table of n, a(n) for n = 1..36

Cf. A000005, A006218, A057494, A095256.

\\ too slow for n > 20; here b(n) is A006218(n).
b(n)={sum(k=1, sqrtint(n), n\k)*2 - sqrtint(n)^2}
a(n)={b(10^n-1)-b(10^(n-1)-1)} \\ Andrew Howroyd, Jan 13 2020

From Andrew Howroyd, Jan 13 2020: (Start)
a(n) = A006218(10^n-1) - A006218(10^(n-1)-1).
a(n) = A057494(n) - A057494(n-1) - 2*n - 1. (End)

Terms a(11) and beyond from Andrew Howroyd, Jan 13 2020

cp's OEIS Frontend

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.

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A153816 a(n) = Sum_{i=1..(10^n-1)/9} floor(((10^n-1)/9)/i).

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A174425 Total number of divisors of all n-digit numbers.

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