[math-fun] Iterating the sum-of-divisors function

Set a_k(0) = k, and a_k(n+1) = sigma(a_k(n)), where sigma(x) is the sum of the divisors of x. My impression is that a_k grows just barely super-exponentially, with local chaos, but very smoothly overall (regardless of k). Does anyone have any intuitions about whether a_2(n) = a_5(m) for any n, m? That is, are the forward orbits of 2 and 5 completely disjoint?