On the natural density of integers n for which σ(kn+r₁) >σ(kn+r₂)

2023/11/01 by Xin-Qi Luo, Luo, Xin-qi, Ren, Chen-kai
Computer Science · Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Graph Labeling and Dimension Problems #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2311.00295

openalex publication_date 2023/11/01 · openalex created_date 2023/11/03 · openalex updated_date 2026/07/28

Abstract

For any positive integer n, let σ(n)=∑_{d| n} d. In 2020, M. Kobayashi and T. Trudgian showed that the natural density of positive integers n with σ(kn+r₁) ≥ σ(kn+r₂) is between 0.053 and 0.055. In this paper, we extend their result. For integers k>r₁>r₂≥ 0, we provide an estimate on the natural density of positive integers n for which σ(kn+r₁) > σ(kn+r₂). We also calculate some special cases with certain k,r₁ and r₂. We also compute explicit bounds for specific k,r₁,r₂ to illustrate the variation of density.

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