A195153 -id:A195153 - cp's OEIS frontend

A195150 Number of divisors d of n such that d-1 does not divide n.

0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 3, 3, 1, 3, 1, 3, 3, 2, 1, 4, 2, 2, 3, 4, 1, 4, 1, 4, 3, 2, 3, 5, 1, 2, 3, 5, 1, 4, 1, 4, 5, 2, 1, 6, 2, 4, 3, 4, 1, 5, 3, 5, 3, 2, 1, 6, 1, 2, 5, 5, 3, 5, 1, 4, 3, 6, 1, 7, 1, 2, 5, 4, 3, 5, 1, 7, 4, 2, 1, 7, 3, 2

Offset: 1

Omar E. Pol, Sep 19 2011

Define "subdivisor" of n to be the positive integer b such that b = d - 1, if d divides n and b does not divide n. For the list of subdivisors of n see A195153.

First occurrence of k is given in A173569. - Robert G. Wilson v, Sep 23 2011

			a(24) = 4 since the divisors of 24 are 1,2,3,4,6,8,12,24, so the subdivisors of 24 are 5,7,11,23 because 6-1 = 5, 8-1 = 7, 12-1 = 11 and 24-1 = 23. Note that the positive integers 1,2,3 are not subdivisors of 24 because they are divisors of 24.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A001620, A027750, A066272, A195153, A137921, A173569.

a195150 n = length [d | d <- [3..n], mod n d == 0, mod n (d-1) /= 0]
-- Reinhard Zumkeller, Sep 23 2011

f[n_] := Module[{d = Divisors[n]}, Length[Select[Rest[d-1], Mod[n, #] > 0 &]]]; Table[f[n], {n, 100}] (* T. D. Noe, Sep 22 2011 *)

a(n) = A137921(n) - 1. - Robert G. Wilson v, Sep 23 2001

Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 3), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 18 2024

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

Original entry on oeis.org

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

Omar E. Pol, Sep 19 2011

First differs from A055874 at a(20).

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

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

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

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

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

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

cp's OEIS Frontend

A195150 Number of divisors d of n such that d-1 does not divide n.

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