On the iterates of arithmetic functions in a class (Q1184823)

For \(\gamma\geq 2\) let \(f^ \gamma=f(f^{\gamma-1})\) be the iterates of an arithmetic function \(f=f^ 1\) and denote by \(S_{f,Q}(\gamma)\) the set of all positive integers \(n\), for which \(f^ \gamma(n)\) is a divisor of \(Qn\). In the present paper the authors completely determine all integers \(\pi\) in the set \(S_{f,Q}(\gamma)\) for odd integers \(Q\) and for functions \(f\) in a certain class of arithmetical functions. Since this class contains Euler's totient function \(\varphi\), a result of \textit{F. Halter-Koch} and \textit{W. Steindl} [Arch. Math. 42, 362-365 (1984; Zbl 0532.10002)] is generalized.