Are All Perfect Numbers Even?/Progress/Prime Factors

Theorem
An odd perfect number has:
 * at least $8$ distinct prime factors
 * at least $11$ distinct prime factors if $3$ is not one of them
 * its greatest prime factor is greater than $1 \, 000 \, 000$
 * its second largest prime factor is greater than $1000$
 * at least one of the prime powers factoring it is greater than $10^{18}$
 * if less than $10^{9118}$ then it is divisible by the $6$th power of some prime.