A330348 a(n) is the number of divisors of n whose last digit equals the last digit of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 1, 3, 2, 2, 1, 2, 3, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 4, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 3, 1, 1, 4, 1, 2, 1, 1, 4, 2, 2, 1, 3, 2, 1, 1, 2, 1

Offset: 1

Author

Bernard Schott, Jun 04 2020

Keywords

• nonn
• base

Comments

Inspired by Project Euler, Problem 474 (see link).

a(n) >= 1.

When n > 10 ends with 0, 1, 2 or 5, then a(n) >= 2.

The first 19 terms are the same as A038769, but a(20) = 2 and A038769(20) = 1.

From Robert Israel, Jun 04 2020: (Start)

a(10*n) = A000005(n).

If n is odd, then a(2*n) = a(n) and a(5*n) = A000005(n). (End)

Integers all of whose divisors end with the same last digit (which is necessarily 1) are in A004615. - Bernard Schott, May 07 2021

Examples

			The divisors of 12 that end in 2 are 2 and 12, so a(12) = 2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Project Euler, Problem 474: Last digits of divisors

Crossrefs

Cf. A000005 (number of divisors), A004615, A010879 (last digit of n), A038769.

Programs

Maple

f:= proc(n) local t;
t:= n mod 10;
nops(select(k -> k mod 10 = t, numtheory:-divisors(n)))
end proc:
map(f, [$1..100]); # Robert Israel, Jun 04 2020

Mathematica

a[n_] := DivisorSum[n, 1 &, Mod[# - n, 10] == 0 &]; Array[a, 100] (* Amiram Eldar, Jun 04 2020 *)

PARI

a(n) = my(u=n%10); sumdiv(n, d, d%10 == u); \\ Michel Marcus, Jun 04 2020

Python

from sympy import divisors
def a(n): return sum((n-d)%10 == 0 for d in divisors(n, generator=True))
print([a(n) for n in range(1, 90)]) # Michael S. Branicky, Aug 15 2022