Odd Spoof Multiperfect Numbers
László Tóth
TL;DR
This work extends the concept of spoof perfect numbers to multiperfect numbers by defining spoof $k$-perfect numbers of two kinds through the relations $\sigma(n)(x+1) = k n x$ and $\sigma(n)(x^2+x+1) = k n x$, with $s = n x$ as the target. A simple, implementable search is used: compute $q = \sigma(n)/(k n)$ and check whether $q$ equals $x/(x+1)$ or $x/(x^2+x+1)$, applying coprimality constraints and delta-based criteria to identify spoof numbers. The study finds no odd first-kind examples beyond Descartes' classical case but reports several odd second-kind examples, including $S = 8999757$ (with spoof factor $61$) and related numbers, alongside a discussion of even spoof numbers. Additionally, the work connects these constructions to Robin's inequality and RH, noting some spoof numbers appear to violate the expected bounds, thereby motivating further computational exploration and generalization to higher multiplicities. The results provide a concrete, scalable framework for discovering and analyzing odd spoof multiperfect numbers and raise questions about their existence for larger $k$ and higher-factor multiplicities.
Abstract
We generalize the definition of spoof perfect numbers to multiperfect numbers and study their characteristics. As a result, we find several new odd spoof multiperfect numbers, akin to Descartes' number. An example is $8999757$, which would be an odd multiperfect number, if one of its prime factors, $61$, was a square. We briefly describe an algorithm for searching for such numbers and discuss a few of their properties.