A Perfect Number Generalization and Some Euclid-Euler Type Results

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 $m\ge 1$. We also provide a characterization of the $\{-1,m\}$-perfect numbers of the form $2^{k}p$ ($k\ge 1$, $p$ an odd prime), as well as a characterization of all even $\{-1,1\}$-perfect numbers.

10 pages