Perfect numbers

The sigma function is multiplicative.
If \(2^k - 1\) is prime, then \(2^{k - 1}\left(2^k - 1\right)\) is perfect.
If \(n\) is an even perfect number, then $$n = 2^{p - 1}\left(2^p - 1\right)$$ for some prime \(p,\) and \(2^p - 1\) is also prime.