The Aliquot Constant, after Bosma and Kane

Let s(n) be the sum of those positive divisors of the natural number n other than n itself. A conjecture of Catalan–Dickson is that the “aliquot” sequence of iterating s starting at any n terminates at 0 or enters a cycle. There is a “counter” conjecture of Guy–Selfridge that while Catalan–Dickson may be correct for most odd numbers n, for most even seeds, the aliquot sequence is unbounded. Lending some support for Catalan–Dickson, Bosma and Kane recently showed that the geometric mean of the numbers s(n)/n for n even tends to a constant smaller than 1. In this paper we reprove their result with a stronger error term and with a finer calculation of the asymptotic geometric mean. In addition we solve the analogous problems for certain subsets of the even numbers, such as the even squarefrees and the multiples of 4.