Generalizing the Abundancy of an Integer

2018/03/28 by David C. Luo, Luo, David C.
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1803.10816

openalex publication_date 2018/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The abundancy index of a positive integer is the ratio between the sum of its divisors and itself. We generalize previous results on abundancy indices by defining a two-variable abundancy index function as I(x,n)\colon\mathbbZ⁺×\mathbbZ⁺→ℚ where I(x,n)=(σ_x(n))/(n^x). Specifically, we extend limiting properties of the abundancy index and construct sufficient conditions for rationals greater than one that fail to be in the image of the function I(x,n).

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