## Involve: A Journal of Mathematics

• Involve
• Volume 6, Number 4 (2013), 493-504.

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

#### 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.

#### Article information

Source
Involve, Volume 6, Number 4 (2013), 493-504.

Dates
Received: 7 December 2012
Revised: 18 May 2013
Accepted: 20 May 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1513733615

Digital Object Identifier
doi:10.2140/involve.2013.6.493

Mathematical Reviews number (MathSciNet)
MR3115982

Zentralblatt MATH identifier
1322.11005

#### Citation

Davis, Nichole; Klyve, Dominic; Kraght, Nicole. On the difference between an integer and the sum of its proper divisors. Involve 6 (2013), no. 4, 493--504. doi:10.2140/involve.2013.6.493. https://projecteuclid.org/euclid.involve/1513733615

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