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\begin{abstract}
Let $n>2$ be a positive integer and let $\phi$ denote Euler's totient function. Define $\phi^1(n)=\phi(n)$ and
$\phi^k(n)=\phi(\phi^{k-1}(n))$ for all integers $k\ge2$.
Define the arithmetic function $S$ by
$S(n)=\phi(n)+\phi^2(n)+\cdots+\phi^c(n)+1$, where $\phi^c(n)=2$.
We say $n$ is a perfect totient number if $S(n)=n$.
We give a list of known perfect totient numbers,
and we give sufficient conditions for the existence of
further perfect totient numbers.
\end{abstract}


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