RO-TST 2014 Problem 11

For every positive integer $n$ , let $\sigma(n)$ denote the sum of all positive divisors of $n$ ( $1$ and $n$ , inclusive). Show that a positive integer $n$ , which has at most two distinct prime factors, satisfies the condition $\sigma(n)=2n-2$ if and only if $n=2^k(2^{k+1}+1)$ , where $k$ is a non-negative integer and $2^{k+1}+1$ is prime.