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

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

Title	iterated totient function
Canonical name	IteratedTotientFunction
Date of creation	2013-03-22 16:33:09
Last modified on	2013-03-22 16:33:09
Owner	CompositeFan (12809)
Last modified by	CompositeFan (12809)
Numerical id	5
Author	CompositeFan (12809)
Entry type	Definition
Classification	msc 11A25
Related topic	PerfectTotientNumber