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

Authors

Nicole Davis
Dominic Klyve, Central Washington UniversityFollow
Nicole Kraght

Department or Administrative Unit

Mathematics

Document Type

Article

Author Copyright

Copyright © 2013 Mathematical Sciences Publishers

Publication Date

2013

Journal

Involve

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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Recommended Citation

Davis, N., Klyve, D. & Kraght, N. (2013). On the difference between an integer and the sum of its proper divisors. Involve, 6(4), 493-504. DOI: 10.2140/involve.2013.6.493