A365680 The number of exponentially squarefree divisors of n.

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 5, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 6, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4

Offset: 1

Author

Amiram Eldar, Sep 15 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A252505 at n = 32.

The number of divisors of n that are exponentially squarefree numbers (A209061), i.e., numbers having only squarefree exponents in their canonical prime factorization.

The sum of these divisors is A365682(n) and the largest of them is A365683(n).

Links

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

Crossrefs

Cf. A000005, A013928, A046100, A209061, A252505, A365682, A365683.

Similar sequences: A325837, A353898.

Programs

Mathematica

f[p_, e_] := Count[Range[e], ?SquareFreeQ] + 1; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

s(n) = sum(k=1, n, issquarefree(k)) + 1;
a(n) = vecprod(apply(x -> s(x), factor(n)[, 2]));

Formula

Multiplicative with a(p^e) = A013928(e+1) + 1.

a(n) <= A000005(n), with equality if and only if n is a biquadratefree number (A046100).