A note on the sum of reciprocals

TL;DR

This paper explores the representation of a fixed positive integer as a sum of reciprocals, with constraints on partial sums representing specific integer combinations based on a given partition.

Contribution

It establishes the existence of a reciprocal sum sequence with partial sums limited to sums over subsets of a given partition, revealing a new structural property.

Findings

Partial sums only represent sums over subsets of the partition

Existence of such reciprocal sequences for any fixed positive integer and partition

Provides a novel connection between partitions and reciprocal sum representations

Abstract

For a fixed positive integer $m$ and any partition $m = m_1 + m_2 + \cdots + m_e$ , there exists a sequence $\{n_{i}\}_{i=1}^{k}$ of positive integers such that $$m=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\cdots+\frac{1}{n_{k}},$$ with the property that partial sums of the series $\{\frac{1}{n_i}\}_{i=1}^{k}$ can only represent the integers with the form $\sum_{i\in I}m_i$, where $I\subset\{1,...,e\}$.