On the denominators of harmonic numbers

TL;DR

This paper investigates the properties of the denominators of harmonic numbers, revealing that for most integers, these denominators are divisible by various least common multiples, with the set of such integers having density one.

Contribution

The paper establishes that the set of positive integers with denominators divisible by the least common multiple of initial segments has density one, advancing understanding of harmonic number denominators.

Findings

The set of n where v_n is divisible by lcm(1,...,⌊n^{1/4}⌋) has density one.

For any positive integer m, the set of n with v_n divisible by m has density one.

v_n is even for all n ≥ 2.

Abstract

Let $H_n$ be the $n$-th harmonic number and let $v_n$ be its denominator. It is well known that $v_n$ is even for every integer $n\ge 2$. In this paper, we study the properties of $v_n$. One of our results is: the set of positive integers $n$ such that $v_n$ is divisible by the least common multiple of $1, 2, \cdots, \lfloor {n^{1/4}}\rfloor $ has density one. In particular, for any positive integer $m$, the set of positive integers $n$ such that $v_n$ is divisible by $m$ has density one.