Families of Values of the Excedent Function $\sigma (n) - 2n$

Abstract

The excedent function, $e(n) := \sigma(n) - 2n$, measures the amount by which the sum of the divisors of an integer exceeds that integer. Despite having been in the mathematical consciousness for more than $2000$ years, there are many unanswered questions concerning the function. Of particular importance to us is the question of explaining and classifying values in the image of $e(n)$ — especially in understanding the ``small'' values. We look at extensive calculated data, and use them as inspiration for new results, generalizing theorems in the literature, to better understand a family of values in this image.