It turns out that these things really are exceptionally hard to discover but a handful of restrictions have been uncovered. Aside from the ones presented in class, here are a few from a wikipedia article:

The smallest prime factor of N is less than (2k + 8) / 3

The largest prime factor of N is greater than 10^8

The second largest prime factor is greater than 10^4, and the third largest prime factor is greater than 100

N has at least 75 prime factors and at least 9 distinct prime factors. If 3 is not one of the factors of N, then N has at least 12 distinct prime factors

An odd perfect number is not divisible by 105

Every odd perfect number is of the form 12m + 1 or 324m + 81 or 468m + 117

This website also has a few links regarding unsolved problems. In the interest of the topic, this includes odd perfect numbers.