Partitions with prescribed sum of reciprocals: asymptotic bounds

2025/02/04 by Wouter van Doorn, van Doorn, Wouter
Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2502.02200

openalex publication_date 2025/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In 1963 Graham proved that every positive integer n ≥ 78 can be written as a sum of distinct positive integers a₁, a₂, …, a_r for which (1)/(a₁) + (1)/(a₂) + … + (1)/(a_r) is equal to 1. In the same paper he managed to further generalize this, and showed that for all positive rationals α and all positive integers m, there exists an n_{α, m} such that every positive integer n ≥ n_{α, m} has a partition with distinct parts, all larger than or equal to m, and such that the sum of reciprocals is equal to α. No attempt was made to estimate the quantity n_{α, m}, however. With n_α := n_{α, 1}, in this paper we provide near-optimal upper bounds on n_α and n_{α, m}, as well as bounds on the cardinality of the set \α: n_α ≤ n\.

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