Nonaliquots and Robbins numbers
Volume 103 / 2005
Colloquium Mathematicum 103 (2005), 27-32
MSC: Primary 11A25; Secondary 11A41, 11N64.
DOI: 10.4064/cm103-1-4
Abstract
Let $\varphi(\cdot)$ and $\sigma(\cdot)$ denote the Euler function and the sum of divisors function, respectively. We give a lower bound for the number of $m\le x$ for which the equation $m=\sigma(n)-n$ has no solution. We also show that the set of positive integers $m$ not of the form $(p-1)/2-\varphi(p-1)$ for some prime number $p$ has a positive lower asymptotic density.
