**Hint:** try to check bellow  papers however they  include iterate sum of divisor function, but you w'ill find solution of your problem , papers include " meta perfect number:" 

**Cohen, Graeme L., and Herman JJ te Riele. "Iterating the sum-of-divisors function."**

**Kátai, I. "On the prime power divisors of the iterates of $\phi(n)$ and $\sigma(n)$, Šiauliai**

**Math." Semin 4.12 (2009): 125-143**
**Experimental Mathematics 5.2 (1996): 91-100**.