On a sum of divisors problem (Q2774634)

The authors discuss a generalization of the notion of perfect number. It concerns a set \(S\) recursively defined by A. Granville: \(1\in S\), and if \(\sum_{d\mid n, d<n, d\in S} d\leq n\), \(n\in N\), \(n>1\), then \(n\in S\). If in this summation formula equality holds instead of inequality, then the number \(n\) is called \(S\)-perfect. The authors find a necessary and sufficient condition for numbers of the form \(2^mp\) (\(p>2\) is a prime) to be \(S\)-perfect. A distribution function for \(S\)-perfect numbers is found as well as an estimate of the density of the set \(S\). These estimations are experimentally verified by the use of a computer. They also give a generalization of \(S_\alpha\) (\(0<\alpha\), \(\alpha\in \mathbb{R}\)) of the set \(S\) and an appropriate notion of \(S_\alpha\)-perfectness. Several interesting open problems and hypotheses concerning sets \(S\) and \(S_\alpha\) are stated.