  In this note, we show that if $N$ is an odd perfect number and $q^{\alpha}$
is some prime power exactly dividing it, then
$\sigma(N/q^{\alpha})/q^{\alpha}>5$. In general, we also show that if
$\sigma(N/q^{\alpha})/q^{\alpha}<K$, where $K$ is any constant, then $N$ is
bounded by some function depending on $K$.
