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

1,16

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.

Links

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

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)).

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A245196 A259362 A303553 * A188585 A294068 A339553

Adjacent sequences: A365547 A365548 A365549 * A365551 A365552 A365553

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 08 2023

Status

approved