**Hint:** try to check [this online book] under this title **The math encyclopedia of smarandach type notion** in number theory include also arithmitic function , And 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**.

[this online book]:https://books.google.dz/books?id=DYEXAgAAQBAJ&pg=PA6&lpg=PA6&dq=best+books+for+sum+divisor+function&source=bl&ots=stuyvM_P2l&sig=ahjZEIrNT4ICHHsw4zGjEveMZZY&hl=fr&sa=X&ved=0ahUKEwicid3or9LYAhWIshQKHQo7CXw4ChDoAQhIMAM#v=onepage&q=best%20books%20for%20sum%20divisor%20function&f=false