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.

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.