  Let $N$ be an odd perfect number. Let $\omega(N)$ be the number of distinct
prime factors of $N$ and let $\Omega(N)$ be the total number of prime factors
of $N$. We prove that if $(3,N)=1$, then $ \frac{302}{113}\omega -
\frac{286}{113} \leq \Omega. $
  If $3|N$, then $\frac{66}{25}\omega-5\leq\Omega.$ This is an improvement on
similar prior results by the author which was an improvement of a result of
Ochem and Rao. We also establish new lower bounds on $\omega(N)$ in terms of
the smallest prime factor of $N$ and establish new lower bounds on $N$ in terms
of its smallest prime factor.
