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

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

Offset: 1

Author

Ctibor O. Zizka, Jan 02 2009

Keywords

• nice
• nonn

Comments

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.

Links

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

Crossrefs

Cf. A000005, A095256, A006218, A002275.

Programs

Mathematica

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

PARI

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

Python

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

Python

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

Formula

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

Extensions

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