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 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.

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