Journal of Integer Sequences, Vol. 10 (2007), Article 07.9.6

Abundancy "Outlaws" of the Form (σ(N) + t)/N


William G. Stanton and Judy A. Holdener
Department of Mathematics
Kenyon College
Gambier, Ohio 43022
USA

Abstract:

The abundancy index of a positive integer $n$ is defined to be the rational number $I(n)=\sigma(n)/n$, where $\sigma$ is the sum of divisors function $\sigma(n)=\sum_{d\vert n}d$. An abundancy outlaw is a rational number greater than 1 that fails to be in the image of of the map $I$. In this paper, we consider rational numbers of the form $(\sigma(N)+t)/N$ and prove that under certain conditions such rationals are abundancy outlaws.

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Received October 25 2006; revised version received August 31 2007; September 25 2007. Published in Journal of Integer Sequences, September 25 2007.

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