#### Vol. 6, No. 4, 2013

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

### Nichole Davis, Dominic Klyve and Nicole Kraght

Vol. 6 (2013), No. 4, 493–504
##### 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 $n<1{0}^{20}$ 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.

##### Keywords
sigma function, sum of divisors, excedents, computational mathematics
##### Mathematical Subject Classification 2010
Primary: 11A25, 11Y70