I just checked how one could implement the search, finding all the divisors for σ(n) can be done with a prime factoring algorithm and then calculating the number of their combinations. But that's just one way.
You can make that more efficient by noting that you need σ(n) to be large, so if you know that, for example, the smallest factor of n is more than ln(n)^4 then you already know this won't be a counter-example. So general purpose factoring doesn't really help in this search, even though, as you say, it's one (naive and simple) approach.
It will work for small n, but it will quickly become infeasible.
