  We study the sums $$ S_f(x) = \sum_{n\leq x}
f\left(\left\lfloor\frac{x}{n}\right\rfloor\right) $$ when $f$ is supported on
$r$th powers with $r\geq 2$. This restriction allows us to give nontrivial
estimates for one of the error terms in the asymptotic expansion of $S_f(x)$.
We also state several conjectures related to our results.
