On the difference between an integer and the sum of its proper divisors

Abstract

Let $\sigma \left(n\right)$ be the sum of the divisors of $n$. Although much attention has been paid to the possible values of $\sigma \left(n\right)-n$ (the sum of proper divisors), comparatively little work has been done on the possible values of $e\left(n\right):=\sigma \left(n\right)-2n$. Here we present some theoretical and computational results on these values. In particular, we exhibit some infinite and possibly infinite families of integers that appear in the image of $e\left(n\right)$. We also find computationally all values of $n<10^{20}$ for which $e\left(n\right)$ is odd, and we present some data from our computations. At the end of this paper, we present some conjectures suggested by our computational work.