A195155 Number of divisors d of n such that d-1 also divides n or d-1 = 0.

1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 3, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 6, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 5, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 5, 1, 2

Offset: 1

Author

Omar E. Pol, Sep 19 2011

Keywords

• nonn
• easy

Comments

First differs from A055874 at a(20).

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000005, A007862, A049820, A055874, A129308, A137921, A195150, A195153.

Programs

Maple

with(numtheory):
a:= n-> add(`if`(d=1 or irem(n, d-1)=0, 1, 0), d=divisors(n)):
seq(a(n), n=1..200);  # Alois P. Heinz, Oct 17 2011

Mathematica

d1[n_]:=Module[{d=Rest[Divisors[n]]},Count[d,?(Divisible[n,#-1]&)]+1]; Array[d1, 90] (* _Harvey P. Dale, Oct 31 2013 *)

Python

from itertools import pairwise
from sympy import divisors
def A195155(n): return 1 if n&1 else 1+sum(1 for a, b in pairwise(divisors(n)) if a+1==b) # Chai Wah Wu, Jun 09 2025

Formula

a(n) = A000005(n) - A195150(n).

a(n) = 1 + A129308(n).

a(2n-1) = 1; a(2n) = 1 + A007862(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Dec 31 2023

a(n) <= A038548(n) <= A000005(n). - Charles R Greathouse IV, Jun 09 2025