Given a positive integer $n$, let $\sigma(n)$ denote the sum of divisors
of $n$. We say that a positive integer $n$ is a *near-perfect* number
if $\sigma(n) - 2n$ is greater than $0$ and a proper divisor of $n$.

**Question:** Is it true that there are only finitely many near-perfect
numbers with more than $4$ distinct prime divisors?

numberof even near-perfect numbers"? Also, your special number has 4 different prime divisors (not more than 4) -- how it's relevant? Also, see oeis.org/A181595 $\endgroup$ – Max Alekseyev Apr 9 '16 at 5:07