Title

On the Difference Between an Integer and the Sum of its Proper Divisors

Document Type

Article

Department or Administrative Unit

Mathematics

Publication Date

2013

Abstract

Let σ(n) be the sum of the divisors of n. Although much attention has been paid to the possible values of σ(n)−n (the sum of proper divisors), comparatively little work has been done on the possible values of e(n):=σ(n)−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(n). We also find computationally all values of n<1020 for which e(n) 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.

Comments

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Journal

Involve

Copyright

Copyright © 2013 Mathematical Sciences Publishers