Odd Spoof Multiperfect Numbers Of Higher Order
László Tóth
TL;DR
The paper extends the spoof multiperfect-number concept to higher-order multiplicities, defining spoof $k$-perfect numbers of order $\alpha$ via $s=nx$ with $\mathcal{S}_{\alpha}=\sum_{a=0}^{\alpha} x^{a}$ and the condition $\sigma(n)\mathcal{S}_{\alpha}=k\,n\,x$. It develops an algorithm to search for such numbers within a wide range, iterating over $n$ and testing a reduced fraction criterion that yields $s=nx$ when satisfied; using this approach, the author reports 14 spoof multiperfect numbers (11 new) with $x$ prime and coprime to $n$, including notable examples like $T=181545$. The work further adapts Robin's inequality to spoof numbers, deriving bounds under the Riemann Hypothesis and yielding a corollary for Descartes-type cases. Overall, the study provides a practical framework to identify odd near-perfect numbers and outlines a path for continued computational discovery. The results highlight that higher-order spoof numbers are accessible and warrant deeper exploration.
Abstract
We extend our previous work on odd spoof multiperfect numbers to the case where spoof factor multiplicities exceed $2$. This leads to the identification of $11$ new integers that would be odd multiperfect numbers if one of their prime factors had higher multiplicity. An example is $181545$, which would be an odd multiperfect number if only one of its prime factors, $3$, had multiplicity $5$.