On the iterates of the sum of unitary divisors (Q1271966)

Let \(\sigma_1(n)=\sigma(n)\) be the sum of all of the divisors of \(n\), and \(\sigma_k(n)=\sigma(\sigma_{k-1}(n))\) for all \(k\geq 2\). \textit{P. Erdős} [Colloq. Math. 17, 195-202 (1967; Zbl 0173.03903)] showed that, for any fixed \(k\geq 2\), we have \(\sigma_k(n)/\sigma_{k-1}(n)\sim k e^\gamma \log\log\log n\) for almost all integers \(n\). The estimate here is dominated by the behaviour of the small prime factors. In this paper the authors consider the sum, \(\sigma^*(n)\), of unitary divisors \(d\) of \(n\), that is divisors for which \((d,n/d)=1\). \textit{P. Erdős} and \textit{M. V. Subbarao} [Lect. Notes Math. 251, 119-125 (1972; Zbl 0228.10033)] showed that \(\sigma^*_k(n)\sim \sigma^*_{k-1}(n)\) for almost all integers \(n\), for \(k=2\), here the small primes playing less of a role. In this paper the authors prove the result for \(k=3\), and hope to extend this to all fixed \(k\) in the future.