On the iterates of some multiplicative functions

Abstract. Let \(f_k(n)\) be the \(k\)-th iterates of a function \(f(n)\), i.e. \(f_0(n):=n\), \(f_1(n):=f(n),\ldots,f_{k+1}(n):=f(f_k(n))~(k=1,2,\cdots)\). We prove that if \(n\in{\Bbb N}\) and the function \(f\) is defined by \(f(p)= p\) and \(f(p^\alpha)=p+p^2\) for all primes \(p\), \(\alpha\ge 2\), then for some \(k\in{\Bbb N}\) there are an $$ u\in\{1,~2,~3,~2.3,~2^3.3^2,~ 2^2.3,~ 2.3^2,~ 2^3.3\} $$ and a square-free \({\mathcal D}\in{\Bbb N}\), \(({\mathcal D},6)=1\) such that \(f_{k}(n)=u.{\mathcal D}\).