A Perfect Number Generalization and Some Euclid-Euler Type Results
Tyler Ross
TL;DR
The paper generalizes the notion of perfect numbers through $S$-perfect numbers, defining first- and second-kind presentations with weights from a fixed set $S$ and connecting to semiperfect and hyperperfect notions. It builds a cohesive framework, explores basic properties, and studies fixed-$S$ families, including key inclusions among $\mathcal{P}(S)$. It establishes infinite families of $\{0,m\}$-perfect and $\{-1,m\}$-perfect numbers for every $m\ge 1$, and provides constructive criteria for numbers of the form $2^k p$ to be $\{-1,m\}$-perfect, as well as a complete characterization of even $\{-1,1\}$-perfect numbers. The work also investigates density and conjectures about odd abundant numbers, notably proposing that every nonsquare odd abundant number might be $\{-1,1\}$-perfect, linking generalizations of perfect numbers to long-standing questions in number theory.
Abstract
In this paper, we introduce a new generalization of the perfect numbers, called $\mathcal{S}$-perfect numbers. Briefly stated, an $\mathcal{S}$-perfect number is an integer equal to a weighted sum of its proper divisors, where the weights are drawn from some fixed set $\mathcal{S}$ of integers. After a short exposition of the definitions and some basic results, we present our preliminary investigations into the $\mathcal{S}$-perfect numbers for various special sets $\mathcal{S}$ of small cardinality. In particular, we show that there are infinitely many $\{0, m\}$-perfect numbers and $\{-1,m\}$-perfect numbers for every $m \geq 1$. We also provide a characterization of the $\{-1,m\}$-perfect numbers of the form $2^kp$ ($k \geq 1$, $p$ an odd prime), as well as a characterization of all even $\{-1, 1\}$-perfect numbers.