A remark on sociable numbers of odd order
Let \(s(n)\) be the sum of the proper divisors of the natural number \(n\). The number \(n\) is called sociable if the sequence \(n,s(n),s(s(n)),\dots\) is purely periodic. In this case the period is called the order of sociability of \(n\). Sociable numbers of order \(1\) are called perfect numbers, and sociable numbers of order \(2\) are called amicable numbers. The main result of the paper shows that the number of sociable numbers \(n\leq x\) of order \(k\) is bounded by the quantity \(x/(\log x)^{1+o(1)}\;(x\to\infty)\) in the case of odd and fixed \(k\).
scientific article; zbMATH DE number 5731631