On the distribution of sociable numbers

For a positive integer n, define s(n) as the sum of the proper divisors of n. If s(n)>0, define s₂(n)=s(s(n)), and so on for higher iterates. Sociable numbers are those n with sk(n)=n for some k, the least such k being the order of n. Such numbers have been of interest since antiquity, when order-1 sociables (perfect numbers) and order-2 sociables (amicable numbers) were studied. In this paper we make progress towards the conjecture that the sociable numbers have asymptotic density 0. We show that the number of sociable numbers in 1,x], whose cycle contains at most k numbers greater than x, is o(x) for each fixed k. In particular, the number of sociable numbers whose cycle is contained entirely in 1,x] is o(x), as is the number of sociable numbers in 1,x] with order at most k. We also prove that but for a set of sociable numbers of asymptotic density 0, all sociable numbers are contained within the set of odd abundant numbers, which has asymptotic density about 1/500.