Partitions with prescribed sum of reciprocals: computational results

TL;DR

This paper investigates the existence of integer partitions with prescribed reciprocal sums, determines key thresholds for such partitions, and characterizes specific sets of rational numbers related to these partitions.

Contribution

It provides exact values of $N_M$ for all $M eq 1$, and characterizes all $eta$ with $n_{eta} \\le 100$, advancing understanding of reciprocal sum partitions.

Findings

Determined $N_M$ for all $M \\ge 2$.

Identified all $eta$ with $n_{eta} \\le 100$.

Abstract

For a positive rational $\alpha$, call a set of distinct positive integers $\{a_1, a_2, \ldots, a_r\}$ an $\alpha$-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 $\alpha$. Define $n_{\alpha}$ to be the smallest positive integer such that for all $n \ge n_{\alpha}$ an $\alpha$-partition of $n$ exists and, for a positive integer $M \ge 2$, define $N_M$ to be the smallest positive integer such that for all $n \ge N_M$ 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 $M \ge 2$, and find the set of all $\alpha$ such that $n_{\alpha} \le 100$.