Partitions with prescribed sum of reciprocals: computational results

2025/02/03 by van Doorn, Wouter
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2502.01409

Abstract

For a positive rational α, call a set of distinct positive integers \a₁, a₂, …, a_r\ an α-partition of n, if the sum of the a_i is equal to n and the sum of the reciprocals of the a_i is equal to α. Define n_α to be the smallest positive integer such that for all n ≥ n_α an α-partition of n exists and, for a positive integer M ≥ 2, define N_M to be the smallest positive integer such that for all n ≥ N_M a 1-partition of n exists where M does not divide any of the a_i. In this paper we determine N_M for all M ≥ 2, and find the set of all α such that n_α ≤ 100.

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