Abstract:

We study the sum

$$F(n) = \sum_{k=1}^{\kappa(n)} \varphi^{(k)}(n) .$$

of consecutive iterations of the Euler function $\varphi(n)$ (where the last iteration satisfies $\varphi^{(\kappa(n))}(n)=1$). We show that for almost all $n$, the difference $\vert F(n) - n\vert$ is not too small, and the ratio $n/F(n)$ is not an integer. The latter result is related to a question about the so-called perfect totient numbers, for which $F(n) = n$.