On some friends of the sociable numbers (Q633271)

Let \(s(n):=\sum_{d\mid n,d<n}d\), \(s_0(n)=n\) and for \(k>0\) define \(s_k(n)=s(s_{k-1}(n))\) if \(s_{k-1}(n)>0\). A social number \(n\) satisfies \(s_k(n)=n\) for some \(k\). Let \(k\geq 1\). The author proves that the set of \(n\) for which \(s_k(n)\) exists and is a multiple of \(n\) has density zero. The set of \(n\) for which \(s_k(n)\) exists and is a divisor of \(n\) has density zero.