On 2-Near Perfect Numbers

Let $\sigma (n)$ be the sum of the positive divisors of $n$. A number $n$ is said to be 2-near perfect if $\sigma (n)=2n+d_1+d_2$, where $d_1$ and $d_2$ are distinct positive divisors of $n$. We give a complete description of those $n$ which are 2-near perfect and of the form $n=2^k{p}^i$ where $p$ is prime and $i\in \{1,2\}$. We also prove related results under the additional restriction where $d_1{d}_2=n$.