Geodesic
Abundancy ``outlaws'' of the form \(\frac{(\sigma(N)+t)}{N}\)
Journal of integer sequences, Tome 10 (2007) no. 9
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.
Classification : 11A25, 11Y55, 11Y70
Keywords: abundancy index, abundancy outlaw, sum of divisors function, perfect numbers 
@article{JIS_2007__10_9_a1,
     author = {Stanton,  William G. and Holdener,  Judy A.},
     title = {Abundancy ``outlaws'' of the form {\(\frac{(\sigma(N)+t)}{N}\)}},
     journal = {Journal of integer sequences},
     year = {2007},
     volume = {10},
     number = {9},
     zbl = {1174.11005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2007__10_9_a1/}
}
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Stanton,  William G.; Holdener,  Judy A. Abundancy ``outlaws'' of the form \(\frac{(\sigma(N)+t)}{N}\). Journal of integer sequences, Tome 10 (2007) no. 9. http://geodesic.mathdoc.fr/item/JIS_2007__10_9_a1/