On the Elementary Symmetric Functions of $\{1,1/2,\dots,1/n\}\backslash\{1/i\}$

TL;DR

This paper investigates the integrality of elementary symmetric functions of specific subsets of harmonic reciprocals, extending previous results by identifying cases where these functions are not integers.

Contribution

It proves that for n ≥ 5, the elementary symmetric functions of the set excluding 1/i are not integers, except for two specific cases, refining earlier findings.

Findings

Elementary symmetric functions are non-integers for n ≥ 5, except for n=i=2 and n=i=4.

Extends previous work by Chen and Tang on the integrality of symmetric functions.

Identifies specific cases where the symmetric functions are integers.

Abstract

In 1946, P. Erd\H{o}s and I. Niven proved that there are only finitely many positive integers $n$ for which one or more of the elementary symmetric functions of $1,1 / 2$, $\cdots, 1 / n$ are integers. In 2012, Y. Chen and M. Tang proved that if $n \geqslant 4$, then none of the elementary symmetric functions of $1,1 / 2, \cdots, 1 / n$ are integers. In this paper, we prove that if $n \geqslant 5$, then none of the elementary symmetric functions of $\{1,1 / 2, \cdots, 1 / n\} \backslash\{1 / i\}$ are integers except for $n=i=2$ and $n=i=4$.