Some problems of Erdős on the sum-of-divisors function (Q2809289)

Let \(\sigma(n)\) denote the sum of all positive divisors of \(n\) and put \(s(n)= \sigma(n)-n\). The authors of this interesting paper reconsider three themes from the late Paul Erdős's works on these functions. First, they refine the known lower and upper bounds for the counting function of numbers \(k\) with \(k\) deficient, but \(s(k)\) abundant, and vice versa. Second, they describe a heuristic argument suggesting the exact asymptotic density of nonaliquot numbers (i.e., numbers not in the range of the function \(s(\cdot)\)). The last part of this paper contains new results on the distribution of the so-called friendly numbers, i.e., numbers with the same values of \(\sigma(n)/n\).