On the total number of prime factors of an odd perfect number

We say $n\in{\mathbb N}$ is perfect if $\sigma(n)=2n$, where $\sigma(n)$denotes the sum of the positive divisors of $n$. No odd perfect numbers are known, but it is well known that if such a number exists, it must have prime factorization of the form $n=p^{\alpha}\prod_{j=1}^{k}q_j^{2\beta_j}$, where $p$, $q_1$, ..., $q_k$ are distinct primes and $p\equiv\alpha\equiv1\pmod4$. We prove that if $\beta_j\equiv1\pmod3$ or $\beta_j\equiv2\pmod5$ for all $j$, $1\le j\le k$, then $3\nmid n$. We also prove as our main result that $\Omega(n)\ge37$, where $\Omega(n)=\alpha+2\sum_{j=1}^{k}\beta_j$. This improves a result of Sayers $( \Omega(n)\ge 29 )$ given in 1986.