Complete characterization of $2$-near perfect numbers with exactly 2 prime factors

Let $\sigma (n)$ be the sum of the positive divisors of $n$. A positive integer $n$ is said to be $2$-near perfect when $\sigma (n)=2n+d_1+d_2$, where $d_1$ and $d_2$ are distinct positive divisors of $n$. We show that there are no odd $2$-near perfect numbers with exactly two prime factors, and that all even $2$-near perfect numbers (i.e. those of the form $2^k{p}^m$, where $p$ is an odd prime) belong to a specific family, provided that $m$ is at least 3. In combination with prior work, these results produce a complete characterization of $2$-near perfect numbers with exactly 2 prime factors.