Some problems of Erdős on the sum-of-divisors function

Abstract:

Let $\sigma (n)$ denote the sum of all of the positive divisors of $n$, and let $s(n) = \sigma (n)-n$ denote the sum of the proper divisors of $n$. The functions $\sigma (\cdot )$ and $s(\cdot )$ were favorite subjects of investigation by the late Paul Erdős. Here we revisit three themes from Erdős’s work on these functions. First, we improve the upper and lower bounds for the counting function of numbers $n$ with $n$ deficient but $s(n)$ abundant, or vice versa. Second, we describe a heuristic argument suggesting the precise asymptotic density of $n$ not in the range of the function $s(\cdot )$; these are the so-called nonaliquot numbers. Finally, we prove new results on the distribution of friendly $k$-sets, where a friendly $k$-set is a collection of $k$ distinct integers which share the same value of $\frac {\sigma (n)}{n}$.