# 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.

Title proof that the sum of the iterated totient function is always odd ProofThatTheSumOfTheIteratedTotientFunctionIsAlwaysOdd 2013-03-22 16:34:26 2013-03-22 16:34:26 PrimeFan (13766) PrimeFan (13766) 6 PrimeFan (13766) Proof msc 11A25