Can anyone help with this problem?
a function is defined to be abundant
The number of integers between 1 and $\displaystyle p^r$ which are not coprime with $\displaystyle p^r$ are $\displaystyle p,\,p^2,\,p^3,\,\ldots,\,p^r.$ Of these, only $\displaystyle p$ is such that $\displaystyle \gcd(p,p^r)=p$ is square-free. So the integers $\displaystyle 1\le a\le p^r$ such that $\displaystyle \gcd(a,p^r)$ is not square-free are the $\displaystyle r-1$ integers $\displaystyle p^2,\,p^3,\,\ldots,\,p^r.$
$\displaystyle \therefore\ F(p^r)=p^r-(r-1)=p^r-r+1.$