A Perfect Number Generalization and Some Euclid-Euler Type Results

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.