A341431 a(n) is the numerator of the asymptotic density of numbers divisible by their last digit in base n.

1, 1, 7, 5, 37, 7, 421, 347, 1177, 671, 14939, 6617, 135451, 140311, 271681, 143327, 5096503, 751279, 91610357, 24080311, 9098461, 830139, 2188298491, 77709491, 925316723, 6609819823, 3567606143, 10876020307, 123417992791, 300151059037, 37903472946337, 32271030591223

Offset

2,3

Links

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

Formula

a(n)/A341432(n) = (1/n) * Sum_{k=1..n-1} gcd(k, n)/k. [corrected by Amiram Eldar, Nov 16 2022]

Example

The sequence of fractions begins with 1/2, 1/2, 7/12, 5/12, 37/60, 7/20, 421/840, 347/840, 1177/2520, 671/2520, 14939/27720, 6617/27720, 135451/360360, 140311/360360, ...
For n=2, the numbers divisible by their last binary digit are the odd numbers (A005408) whose density is 1/2, therefore a(2) = 1.
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) = 1.
For n=10, the numbers divisible by their last digit in base 10 are A034709 whose density is 1177/2520, therefore a(10) = 1177.

Mathematica

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

Crossrefs

Cf. A005408, A034709, A047236, A341432 (denominators).

Sequence in context: A344917 A328758 A070426 * A142883 A146382 A376083

Adjacent sequences: A341428 A341429 A341430 * A341432 A341433 A341434

Keyword

nonn,base,frac,easy

Author

Amiram Eldar, Feb 11 2021

Status

approved