The iterated totient function $\phi^k(n)$ is $a_k$ in the recurrence relation $a_0 = n$ and $a_i = \phi(a_{i - 1})$ for $i > 0$ where $\phi(x)$ is Euler's totient function.

After enough iterations, the function eventually hits 2 followed by an infinite trail of ones. Ianucci et al define the ``class'' $c$ of $n$ as the integer such that $\phi^c(n) = 2$

When the iterated totient function is summed thus: $$\sum_{i = 1}^{c + 1} \phi^i(n)$$ it can be observed that just as $2^x$ is a quasiperfect number when it comes to adding up proper divisors, it is also ``quasiperfect'' when adding up the iterated totient function. Quite unlike regular perfect numbers, $3^x$ (which are obviously odd) are ``perfect'' when adding up the iterated totient.

Bibliography

1

D. E. Ianucci, D. Moujie & G. L. Cohen, ``On Perfect Totient Numbers'' Journal of Integer Sequences, 6, 2003: 03.4.5

2

R. K. Guy, Unsolved Problems in Number Theory New York: Springer-Verlag 2004: B42