Question 6
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.