The index of composition of the iterates of the Euler function

Abstract: The index of composition of an integer $n\ge 2$ is defined as $\lambda(n) = (\log n)/(\log \gamma(n))$, where $\gamma(n)$ stands for the largest square-free divisor of $n$. Let $\varphi$ stand for the Euler totient function. We show that the index of composition of the $k$-fold iterate of $\varphi(n)$ is 1 on a set of density 1 and that an analogous result holds if $n$ runs over the set of shifted primes.

Keywords: index of composition, Euler function, shifted primes

Classification (MSC2000): 11N37; 11N64, 11K65, 11N36

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