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

Davis, Nichole; Klyve, Dominic; Kraght, Nicole

doi:10.2140/involve.2013.6.493

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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

DOI: 10.2140/involve.2013.6.493

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 $e (n) : = σ (n) - 2 n$ . 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 $n < 1 0^{20}$ 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.

Keywords

sigma function, sum of divisors, excedents, computational mathematics

Mathematical Subject Classification 2010

Primary: 11A25, 11Y70

Milestones

Received: 7 December 2012

Revised: 18 May 2013

Accepted: 20 May 2013

Published: 8 October 2013

Communicated by Kenneth S. Berenhaut

Authors

Nichole Davis
13561 Macadam Road, South Tukwila, WA 98168 United States

Dominic Klyve
2400 North Ellington Street Ellenburg, WA 98926 United States

Nicole Kraght
14235 SE 224th Street Kent, WA 98042 United States

Vol. 6, No. 4, 2013