On the denominators of $\varepsilon$-harmonic numbers

Bull. Korean Math. Soc. Published online September 1, 2020

Bing-Ling Wu and Xiao-Hui Yan
Changzhou University, Anhui Normal University

Abstract : Let $H_n$ be the $n$-th harmonic number and let $v_n$ be its denominator. Shiu has proved that there are infinite many positive integers $n$ with $v_n =v_{n+1}$. Recently, Wu and Chen have proved that the set of positive integers $n$ with $v_n = v_{n+1}$ has density one. They also have proved that the same result is true for the denominators of alternating harmonic numbers. In this paper, we prove that the result is true for the denominators of $\varepsilon$-harmonic numbers, where $\varepsilon=\{ \varepsilon_i \}_{i=1}^\infty $ is a pure recurring sequence with $\varepsilon_i\in \{-1,1\}$.