2007/07/15 by Lourdes Benito, Benito, Lourdes
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematics and Applications #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.0707.2190
arxiv created 2007/07/15 · openalex publication_date 2007/07/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For n≤ 1.5 ⋅ 1010, we have found a total number of 1268 solutions to the Erdös-Sierpiński problem finding positive integer solutions of σ(n)=σ(n+1), where σ(n) is the sum of the positive divisors of n. On the basis of that set of solutions the following empirical properties are enunciated: first, all the σ(n), n being a solution, are divisible by 6; second, the repetition of solutions leads to the formulation of a new problem: Find the natural numbers n such that σ(n)=σ(n+1)=σ(n+k)=σ(n+k+1) for some positive integer k. A third empirical property concerns the asymptotic behavior of the function of n that gives the number of solutions for m less or equal to n, which we find to be as n1/3. Finally some theorems related to the Erdös-Sierpiński problem are enunciated and proved.