Integers representable as the sum of powers of their prime factors (Q873790)

From the text: Given an integer \(\alpha \geq 2\), let \(S_\alpha\) be the set of those positive integers \(n\) with at least two distinct prime factors, which can be written as \(n=\sum_{p\mid n}p^\alpha\). We obtain general results concerning the nature of the sets \(S_\alpha\) and we also identify all those \(n\in S_3\) which have exactly three prime factors. Theorem 1: If \(n\in S_3\) and \(\omega(n)=3\), then \(n=2\cdot3^3\cdot 7\) or \(n=2^2\cdot 7^2\cdot 13\). We then consider the set \(T\) (resp. \(T_0\)) of those positive integers \(n\), with at least two distinct prime factors, which can be written as \(n=\sum_{p\mid n}p^{\alpha_p}\), where the exponents \(\alpha_p \geq 1\) (resp. \(\alpha_p\geq 0\)) are allowed to vary with each prime factor \(p\). We examine the size of \(T(x)\) (resp. \(T_0(x)\)), the number of positive integers \(n\leq x\) belonging to \(T\) (resp. \(T_0\)). Theorem 2: As \(x\to\infty\), we have \[ T_0(x)\leq x\exp\left\{-(1+o(1))\sqrt{\tfrac 16\log x\log\log x}\right\}. \] Although we cannot prove that \(T\) is an infinite set, a heuristic argument shows that \[ \exp\left(\frac 2e (1+o(1))\frac{\log x}{(\log\log x)^2}\right)\leq T(x)\leq x^{1/2+o(1)}. \]