On asymptotic properties of aliquot sequences

Abstract:

Put ${s^{(1)}}(n) = \sigma (n) - n,\sigma (n) = {\Sigma _{d/n}}d$. ${s^k}(n) = {s^{(1)}}({s^{(k - 1)}}(n))$. In this note we prove that for every k the density of integers satisfying \[ {s^k}(n) = (1 + \sigma (1))n{((\sigma (n) - n)/n)^k}\] is 1. Several unsolved problems are stated.