Illinois Journal of Mathematics

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

Paul Pollack

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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 $O(x/\log{x})$.

Article information

Illinois J. Math., Volume 58, Number 1 (2014), 125-147.

First available in Project Euclid: 1 April 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11A25: Arithmetic functions; related numbers; inversion formulas
Secondary: 11N37: Asymptotic results on arithmetic functions 11N60: Distribution functions associated with additive and positive multiplicative functions


Pollack, Paul. Some arithmetic properties of the sum of proper divisors and the sum of prime divisors. Illinois J. Math. 58 (2014), no. 1, 125--147. doi:10.1215/ijm/1427897171.

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