  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 $\sim10 \log(k)$.
