iterated totient function

The iterated totient function ${\varphi }^k(n)$ is $a_k$ in the recurrence relation $a_0=n$ and $a_i=\varphi (a_{i-1})$ for $i>0$, where $\varphi (x)$ 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 ${\varphi }^c(n)=2$.

When the iterated totient function is summed thus:

$$\sum _{i=1}^{c+1}{\varphi }^i(n)$$

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.