2025/12/04 by Ross, Tyler
Mathematics · #11A25 (Primary) #11A67 #11Y55 (Secondary) #Advanced Mathematical Identities #Advanced Mathematical Theories #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2512.04417
openalex publication_date 2025/12/04 · openalex created_date 2025/12/06 · openalex updated_date 2026/07/28
In this paper, we introduce a new generalization of the perfect numbers, called S-perfect numbers. Briefly stated, an S-perfect number is an integer equal to a weighted sum of its proper divisors, where the weights are drawn from some fixed set S of integers. After a short exposition of the definitions and some basic results, we present our preliminary investigations into the S-perfect numbers for various special sets 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 ≥ 1. We also provide a characterization of the \-1,m\-perfect numbers of the form 2kp (k ≥ 1, p an odd prime), as well as a characterization of all even \-1, 1\-perfect numbers.