A386470 The number of divisors of n whose exponents in their prime factorization are squares.

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

Offset: 1

Author

Amiram Eldar, Jul 22 2025

Keywords

• nonn
• easy
• mult

Comments

First differs from A365171 and A369310 at n = 32.

First differs from A365488 at n = 128.

The number of terms in A197680 that divide n.

The sum of these divisors is A386471(n) and the largest of them is A386469(n).

Links

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

Crossrefs

Cf. A000005, A005117, A197680, A365171, A365488, A369310, A386469, A386471.

Programs

Mathematica

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

PARI

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

Formula

Multiplicative with a(p^e) = A048760(e) + 1.

a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).