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.
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 |