Density results on hyperharmonic integers

Abstract

It was conjectured that there are no hyperharmonic integers $h_n^{(r)}$ except 1. In 2020, a disproof of this conjecture was given by showing the existence of infinitely many hyperharmonic integers. However, the corresponding proof does not give any general density results related to hyperharmonic integers. In this paper, we first get better error estimates for the counting function of the pairs $(n,r)$ that correspond to non-integer hyperharmonic numbers using sums on gaps between consecutive prime numbers. Then, based on a plausible assumption on prime powers with restricted digits, we show that there exist positive integers $n$ such that the set of positive integers $r$ where $h_n^{(r)}\in \mathbb{Z}$ has positive density. Apart from that, we also obtain exact densities of sets $\{r\in {\mathbb{Z}}_{>0}:h_{33}^{(r)}\in \mathbb{Z}\}$ and $\{r\in {\mathbb{Z}}_{>0}:h_{39}^{(r)}\in \mathbb{Z}\}$. Finally, we give the smallest hyperharmonic integer $h_n^{(r)}$ greater than 1, which is obtained when $n=33$ and $r=10667968$.