Bounds for Odd $k$-Perfect Numbers

Let $k\ge 2$ be an integer. A natural number $n$ is called $k$-perfect if $\sigma (n)=kn\mathrm{.}$ For any integer $r\ge 1$ we prove that the number of odd $k$-perfect numbers with at most $r$ distinct prime factors is bounded by $k{4}^{r^3}$.