A372380 The number of divisors of n that are numbers whose number of divisors is a power of 2.

1, 2, 2, 2, 2, 4, 2, 3, 2, 4, 2, 4, 2, 4, 4, 3, 2, 4, 2, 4, 4, 4, 2, 6, 2, 4, 3, 4, 2, 8, 2, 3, 4, 4, 4, 4, 2, 4, 4, 6, 2, 8, 2, 4, 4, 4, 2, 6, 2, 4, 4, 4, 2, 6, 4, 6, 4, 4, 2, 8, 2, 4, 4, 3, 4, 8, 2, 4, 4, 8, 2, 6, 2, 4, 4, 4, 4, 8, 2, 6, 3, 4, 2, 8, 4, 4, 4

Offset: 1

Author

Amiram Eldar, Apr 29 2024

Keywords

• nonn
• easy
• mult

Comments

First differs from A061389 and A322483 at n = 32.

First differs from A380922 at n = 128. - Vaclav Kotesovec, Apr 22 2025

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A005117, A036537, A372379.

Cf. A061389, A322483, A380922.

Programs

Mathematica

f[p_, e_] := Floor[Log2[e + 1]] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = vecprod(apply(x -> exponent(x+1)+1, factor(n)[, 2]));

Python

from math import prod
from sympy import factorint
def A372380(n): return prod((e+1).bit_length() for e in factorint(n).values()) # Chai Wah Wu, Apr 30 2024

Formula

Multiplicative with a(p^e) = floor(log_2(e+1)) + 1.

a(n) = A000005(n) if and only if n is squarefree (A005117).