iterated totient function

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.

References

Title iterated totient function IteratedTotientFunction 2013-03-22 16:33:09 2013-03-22 16:33:09 CompositeFan (12809) CompositeFan (12809) 5 CompositeFan (12809) Definition msc 11A25 PerfectTotientNumber