The denominators of harmonic numbers (Revised)

TL;DR

This paper investigates the behavior of denominators of harmonic numbers, their relation to least common multiples, and establishes density results and conditions involving prime factors.

Contribution

It proves the existence of infinitely many n where the denominators of harmonic numbers relate to the LCM of 1 to n, and explores prime factor conditions affecting this relationship.

Findings

Denominators of harmonic numbers do not increase monotonically.

The set of n where p times the denominator divides the LCM has positive harmonic density.

Existence of n satisfying divisibility conditions involving multiple primes and harmonic number denominators.

Abstract

The denominators $d_n$ of the harmonic number $1+\frac12+\frac13+\cdots+\frac1n$ do not increase monotonically with~$n$. It is conjectured that $d_n=D_n={\rm LCM}(1,2,\ldots,n)$ infinitely often. For an odd prime $p$, the set $\{n:pd_n|D_n\}$ has a harmonic density. Moreover, for $2<p_1<p_2<\cdots<p_k$, with $\log p_1/\log p_i$ ($1\le i\le k$) being linearly independent, there exists $n$ such that $p_1p_2\cdots p_kd_n|D_n$.