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

2,1

Comments

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

Links

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

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.

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A025527 A334958 A205957 * A092144 A224497 A305753

Adjacent sequences: A341429 A341430 A341431 * A341433 A341434 A341435

Keyword

nonn,base,frac,easy

Author

Amiram Eldar, Feb 11 2021

Status

approved