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Homeproof that the sum of the iterated totient function is always odd

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# proof that the sum of the iterated totient function is always odd

Given a positive integer $n$, it is always the case that

$2\not|\sum_{{i=1}}^{{c+1}}\phi^{i}(n),$ |

where $\phi^{i}(x)$ is the iterated totient function and $c$ is the integer such that $\phi^{c}(n)=2$.

Accepting as proven that $n>\phi(n)$ and $2|\phi(n)$ for $n>2$, it is clear that summing up the iterates of the totient function up to $c$ is summing up a series of even numbers in descending order and that this sum is therefore itself even. Then, when we add the $c+1$ iterate, the sum turns odd.

As a bonus, this proves that no even number can be a perfect totient number.

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## Mathematics Subject Classification

11A25*no label found*

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