On the Density of Spoof Odd Perfect Numbers

TL;DR

This paper investigates the properties and density of odd integers related to spoof odd perfect numbers, providing bounds on their factors and conjecturing their distribution follows a logarithmic pattern.

Contribution

It introduces new bounds on the factors of spoof odd perfect numbers and proposes a conjecture on the asymptotic density of related integers.

Findings

If $D=pq$ is a spoof odd perfect number, then $q>10^{12}$.

Irregular digit patterns are observed in the set $\\mathcal{S}$.

The density of such numbers below $k$ is conjectured to be approximately $10 \\log(k)$.

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 $\sim10 \log(k)$.