PlanetMath: iterated totient function

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iterated totient function

(Definition)

The iterated totient function $\phi^k(n)$ is in the recurrence relation and $a_i = \phi(a_{i - 1})$ for , 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” of as the integer such that $\phi^c(n) = 2$ .

When the iterated totient function is summed thus:

$\displaystyle \sum_{i = 1}^{c + 1} \phi^i(n)$

it can be observed that just as

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,

(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

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