A365550 The number of square coreful divisors of n.

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Amiram Eldar, Sep 08 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A188585 at n = 64.

A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n.

Examples

			a(16) = 2 since the coreful divisors of 16 are {2, 4, 8, 16}, and 2 of them, 4 and 16, are squares.
		

Links

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

Crossrefs

Cf. A001694, A005361 (number of coreful divisors), A046951 (number of square divisors), A325837.

Programs

Mathematica

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

PARI

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

Formula

Multiplicative with a(p^e) = floor(e/2).

a(n) > 0 if and only if n is a powerful number (A001694).

Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 - 1/p^s + 1/p^(3*s)).