A383762 The number of unitary divisors of n that are exponentially squarefree numbers.

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 4, 1, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 2, 4, 2, 8, 2, 2, 4, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 2, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 8, 2, 4, 4, 2, 4, 8, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 4, 8, 2, 2, 1, 4, 2, 8, 4, 4, 4

Offset: 1

Author

Amiram Eldar, May 09 2025

Keywords

• nonn
• easy
• mult

Comments

First differs from A365499 at n = 32.

The sum of these divisors is A383763(n) and the largest of them is A383764(n).

Links

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

Crossrefs

Cf. A005117, A034444, A077610, A209061, A365499, A365680, A383763, A383764.

Programs

Mathematica

f[p_, e_] := If[SquareFreeQ[e], 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

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

Formula

Multiplicative with a(p^e) = 2 if e is squarefree (A005117), and 1 otherwise.

a(n) <= A034444(n), with equality if and only if n is an exponentially squarefree number (A209061).

a(n) <= A365680(n), with equality if and only if n is a squarefree number.