The Non-Euler Part of a Spoof Odd Perfect Number is Not Almost Perfect

TL;DR

This paper investigates spoof odd perfect numbers, also known as Descartes numbers, and demonstrates that the component $k$ in their factorization cannot be almost perfect, extending properties known for odd perfect numbers.

Contribution

It establishes that in spoof odd perfect numbers, the factor $k$ is definitively not almost perfect, providing new insights into their structure.

Findings

$k$ is not almost perfect in spoof odd perfect numbers

Results analogous to odd perfect numbers are established for spoof numbers

Extends understanding of the divisor sum properties in non-Euler parts

Abstract

We call $n$ a spoof odd perfect number if $n$ is odd and $n=km$ for two integers $k,m>1$ such that $\sigma(k)(m+1)=2n$, where $\sigma$ is the sum-of-divisors function. In this paper, we show how results analogous to those of odd perfect numbers could be established for spoof odd perfect numbers (otherwise known in the literature as Descartes numbers). In particular, we prove that $k$ is not almost perfect.