On functions arising from generalized Euler functions (Q1095172)

Denote by \(\varphi^ k\) the \(k\)-th iterate of Euler's totient function \(\varphi\) and by \(\varepsilon\) the characteristic function of the set of even integers. Define the arithmetical function \(n\mapsto c(n)\) by \(c(n)=k(n)+\varepsilon (n)\) where \(k(1)=k(2)=0\) and the relation \(\varphi^{k(n)}(n)=2\) determines \(k(n)\) uniquely for \(n\geq 3\). \textit{S. S. Pillai} [Bull. Am. Math. Soc. 35, 837--841 (1929; JFM 55.0107.02)] and \textit{H. N. Shapiro} [Am. Math. Mon. 50, 18--30 (1943; Zbl 0061.08002)] independently obtained the bounds \[ \frac{\log (n/2)}{\log 3} < c(n) < \frac{\log n}{\log 2} \] and in another paper [Commun. Pure Appl. Math. 3, 259--272 (1950; Zbl 0039.27306)] \textit{H. N. Shapiro} studied some properties of the iterates of a certain class \(K\) of arithmetical functions including \(\varphi\). The present authors extend the Pillai-Shapiro result on \(c(n)\) to a subclass of \(K\), thereby achieving a slightly sharper bound in the case of \(\varphi (n)\).