Any integer
![$n\ge 2$](abs/img1.gif)
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
![$\pow(n)$](abs/img2.gif)
and
![$\sq(n)$](abs/img3.gif)
respectively, setting for convenience
![$\pow(1)=\sq(1)=1$](abs/img4.gif)
.
We first
examine the behavior of the counting functions
![$\sum_{n\le x,\,
{\scriptsize\sq}(n)\le y} 1$](abs/img5.gif)
and
![$\sum_{n\le x,\, {\scriptsize
\pow}(n)\le y} 1$](abs/img6.gif)
.
Letting
P(
n) stand for the largest prime factor of
n, we then provide asymptotic values of
![$A_y(x):=\sum_{n\le x,\,
P(n)\le y} \pow(n)$](abs/img7.gif)
and
![$B_y(x) :=\sum_{n\le x,\, P(n)\le y} \sq(n)$](abs/img8.gif)
when
y=
x1/u with
![$u\ge 1$](abs/img9.gif)
fixed. We also examine the size of
Ay(
x) and
By(
x) when
![$y=(\log x)^\eta$](abs/img10.gif)
for some
![$\eta>1$](abs/img11.gif)
.
Finally, we prove that
Ay(
x) will coincide with
By(
x) in the
sense that
![$\log(A_y(x)/x) = (1+o(1))\log(B_y(x)/x)$](abs/img12.gif)
as
![$x\to \infty$](abs/img13.gif)
if we choose
![$y=2\log x$](abs/img14.gif)
.