The index of composition of the iterates of the Euler function

De Koninck, Jean-Marie and Kátai, Imre (2016) The index of composition of the iterates of the Euler function. ACTA MATHEMATICA ACADEMIAE PAEDAGOGICAE NYÍREGYHÁZIENSIS, 32 (2). pp. 303-311. ISSN 0866-0174

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Abstract

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

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