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Let f W N ! N 0 be a multiplicative arithmetic function such that for all primes p and positive integers˛, f .p˛/ < p˛and f .p/jf .p˛/. Suppose also that any prime that divides f .p˛/ also divides pf .p/. Define f .0/ D 0, and let H.n/ D lim m!1 f m .n/, where f m denotes the m t h iterate of f . We prove that the function H is completely multiplicative.doi:10.18514/mmn.2015.1448 fatcat:kmw6lgtz4ncx3pqt7f553wuyhq