Integers representable as the sum of powers of their prime factors

Abstract

Given an integer $\alpha \geqslant 2$, let $S_\alpha$ be the set of those positive integers $n$, with at least two distinct prime factors, which can be written as $n = \sum\limits_{p|n}p^\alpha$. 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. We then consider the set $T \ (\mathrm{resp.} \ T_{0})$ of those positive integers $n$, with at least two distinct prime factors, which can be written as $n = \sum\limits_{p|n}p^{\alpha_{p}}$, where the exponents $\alpha_{p} \geqslant 1 \ (\mathrm{resp}. \ \alpha_{p} \geqslant 0)$ are allowed to vary with each prime factor $p$. We examine the size of $T(x) \ (\mathrm{resp}. \ T_{0}(x))$, the number of positive integers $n \leqslant x$ belonging to $T \ (\mathrm{resp}. \ T_{0})$.