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

TL;DR

This paper determines when the symmetric sums $S(n,i,k)$, the $k$-th elementary symmetric function of $igl\{1,1/2,\dots,1/n\bigr\}$ with the term $1/i$ removed, can be integers. It combines a prime‑availability argument leveraging a lower bound on primes in $(n/(k+3),n/(k+1)]$ with bounds on the theta function to rule out integrality for large $n$, and uses a computer‑assisted, recursive enumeration for small $n$ to exhaust all cases. The main result is that for $n\ge 5$, none of the $S(n,i,k)$ are integers except for the two exceptional triples $(n,i,k)=(2,2,1)$ and $(4,4,2)$. This adds to the Erdős–Niven and Chen–Tang line of work by giving a precise classification in the removal‑one‑term setting and clarifying the integrality landscape of harmonic‑type symmetric sums.

Abstract

In 1946, P. Erdős 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 $n=i=4$.

Theorems & Definitions (7)

• Theorem 1
• Lemma 2: see 9
• Lemma 3
• proof
• Lemma 4
• proof
• proof : Proof of Theorem \ref{['thm:main']}