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Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 32, No. 2, pp. 303-311 (2016)
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The index of composition of the iterates of the Euler function

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Jean-Marie De Koninck and Imre Kátai

Université Laval and Eötvös Loránd University

**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

**Full text of the article:**

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© 2016
FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition
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