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

Nichole Davis, Dominic Klyve, and Nicole Kraght

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11A25: Arithmetic functions; related numbers; inversion formulas 11Y70: Values of arithmetic functions; tables

sigma function sum of divisors excedents computational mathematics


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.

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