A341432 a(n) is the denominator of the asymptotic density of numbers divisible by their last digit in base n.

2, 2, 12, 12, 60, 20, 840, 840, 2520, 2520, 27720, 27720, 360360, 360360, 720720, 720720, 12252240, 4084080, 232792560, 77597520, 33256080, 5173168, 5354228880, 356948592, 3824449200, 26771144400, 11473347600, 80313433200, 332727080400, 2329089562800, 144403552893600

Offset: 2

Author

Amiram Eldar, Feb 11 2021

Keywords

• nonn
• base
• frac
• easy

Comments

a(n) divides A003418(n), and a(n) = A003418(n) for n = 1, 2, 4, 6, 8, 10, 12, ...

Examples

			For n=2, the numbers divisible by their last binary digit are the odd numbers (A005408) whose density is 1/2, therefore a(2) = 2.
For n=3, the numbers divisible by their last digit in base 3 are the numbers that are congruent to {1, 2, 4} mod 6 (A047236) whose density is 1/2, therefore a(3) = 2.
For n=10, the numbers divisible by their last digit in base 10 are A034709 whose density is 1177/2520, therefore a(10) = 2520.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 2..2300

Crossrefs

Cf. A003418, A005408, A034709, A047236, A185399, A341431 (numerators).

Programs

Mathematica

a[n_] := Denominator[Sum[GCD[k, n]/k, {k, 1, n - 1}]/n]; Array[a, 32, 2]

Formula

A341431(n)/a(n) = (1/n) * Sum_{k=1..n-1} gcd(k, n)/k. [corrected by Amiram Eldar, Nov 16 2022]

a(prime(n)) = A185399(n), for n > 1.