Journal of Integer Sequences, Vol. 17 (2014), Article 14.8.6 |

Département de mathématiques et de statistique

Université Laval

Québec G1V 0A6

Canada

**Abstract:**

Any integer
can be written in a unique way as the product of
its powerful part and its squarefree part, that is, as *n*=*mr* where *m* is a powerful number and *r* a squarefree number, with gcd(*m*,*r*)=1. We
denote these two parts of an integer *n* by
and respectively, setting for convenience
.
We first
examine the behavior of the counting functions
and
.
Letting *P*(*n*) stand for the largest prime factor of
*n*, we then provide asymptotic values of
and
when *y*=*x*^{1/u} with
fixed. We also examine the size of
*A*_{y}(*x*) and *B*_{y}(*x*) when
for some .
Finally, we prove that *A*_{y}(*x*) will coincide with *B*_{y}(*x*) in the
sense that
as
if we choose .

Received July 8 2014;
revised version received July 31 2014.
Published in *Journal of Integer Sequences*, August 5 2014.

Return to