On the distribution of sociable numbers
Let \(s_1(n)= \sigma(n)- n\), the sum of the proper divisors of \(n\), and define \(s_j(n)= s(s_{j-1}(n))\) for \(j\geq 2\) and \(s_{j-1}(n)> 0\). If \(s_k(n)= n\) for some \(k\), then \(n\) is a sociable number and the least such \(k> 0\) is its order. The sociable numbers of order 1 are the perfect numbers and those with order 2 give amicable pairs, the existence of both of which was known in antiquity. The aim of this paper is to make progress towards the conjecture that the set of sociable numbers has asymptotic density zero. The authors establish some ten theorems, in most of which they estimate the number of sociable numbers \(n\leq x\) with a specific property; thus they show that this number is \(o(x)\) when one of the following conditions holds: (1) For fixed \(k\), including \(k= 0\), at most \(k\) numbers \(s_j(n)\) exceed \(x\); (2) The order of \(n\) is at most \(k\); (3) \(n\) is deficient, i.e. \(s(n)< n\); (4) \(n\) is even and abundant, i.e. \(s(n)> n\). It follows that the set of odd abundant numbers, the density of which is shown to be approximately \({1\over 500}\), contains all sociable numbers except for a set of density 0. The relevant work of many mathematicians, including that of Erdős, is described in the introduction with key papers included in the long and useful list of references.
scientific article; zbMATH DE number 5569037
scientific article