## Illinois Journal of Mathematics

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

Paul Pollack

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

#### Article information

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

Dates
First available in Project Euclid: 1 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1427897171

Digital Object Identifier
doi:10.1215/ijm/1427897171

Mathematical Reviews number (MathSciNet)
MR3331844

Zentralblatt MATH identifier
1368.11003

#### Citation

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. https://projecteuclid.org/euclid.ijm/1427897171

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