2025/04/08 by Junnosuke Koizumi, Koizumi, Junnosuke
Computer Science · Mathematics · #11D68 #11J72 #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2504.05933
openalex publication_date 2025/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that sequences of positive integers whose ratios an2/an+1 lie within a specific range are almost uniquely determined by their reciprocal sums. For instance, the Sylvester sequence is uniquely characterized as the only sequence with an2/an+1∈ [2/3,4/3] whose reciprocal sum is equal to 1. This result has applications to irrationality problems. We prove that for almost every real number α> 1, sequences asymptotic to α2n have irrational reciprocal sums. Furthermore, our observations provide heuristic insight into an open problem by Erdős and Graham.