On the Density of Spoof Odd Perfect Numbers

Abstract

We study the set $\mathcal{S}$ of odd positive integers $n$ with the property $2n/\sigma (n)-1=1/x$, for positive integer $x$, i.e., the set that relates to odd perfect and odd "spoof perfect" numbers. As a consequence, we find that if $D=pq$ denotes a spoof odd perfect number other than Descartes' example, with pseudo-prime factor $p$, then $q>10^{12}$. Furthermore, we find irregularities in the ending digits of integers $n\in \mathcal{S}$ and study aspects of its density, leading us to conjecture that the amount of numbers in $\mathcal{S}$ below $k$ is $\sim 10\log (k)$.

Added a note excluding Descartes' example from the Theorem