Some arithmetic properties of the sum of proper divisors and the sum of prime divisors

Abstract

For each positive integer $n$, let $s(n)$ denote the sum of the proper divisors of $n$. If $s(n)>0$, put $s_{2}(n)=s(s(n))$, and define the higher iterates $s_{k}(n)$ similarly. In 1976, Erdős proved the following theorem: For each $\delta>0$ and each integer $K\geq2$, we have

\[-\delta<\frac{s_{k+1}(n)}{s_{k}(n)}-\frac{s(n)}{n}\] for all $1\leq k<K$, except for a set of $n$ of asymptotic density zero. He also conjectured that

\[\frac{s_{k+1}(n)}{s_{k}(n)}-\frac{s(n)}{n}<\delta\] for all $1\leq k<K$ and all $n$ outside of a set of density zero. This conjecture has proved rather recalcitrant and is known only when $K=2$, a 1990 result of Erdős, Granville, Pomerance, and Spiro. We reprove their theorem in quantitative form, by what seems to be a simpler and more transparent argument.

Similar techniques are used to investigate the arithmetic properties of the sum of the distinct prime divisors of $n$, which we denote by $\beta(n)$. We show that for a randomly chosen $n$, the integer $\beta(n)$ is squarefree with the same probability as $n$ itself. We also prove the same result with “squarefree” replaced by “abundant”.

Finally, we prove that for either of the functions $f(n)=s(n)$ or $f(n)=\beta(n)$, the number of $n\le x$ for which $f(n)$ is prime is $O(x/\log{x})$.