Bounds for Odd $k$-Perfect Numbers
  Let $k\ge2$ be an integer. A natural number $n$ is called $k$-perfect if $\sigma(n)=kn.$ For any integer $r\ge1$ we prove that the number of odd $k$-perfect numbers with at most $r$ distinct prime factors is bounded by $k4^{r^3}$. 
Shi-Chao Chen and Hao Luo
math.NT