Bulletin of the
Korean Mathematical Society BKMS

Let $H_n$ be the $n$-th harmonic number and let $v_n$ be its denominator. Shiu proved that there are infinitely many positive integers $n$ with $v_n =v_{n+1}$. Recently, Wu and Chen proved that the set of positive integers $n$ with $v_n = v_{n+1}$ has density one. They also 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 $\varepsilon_i\in \{-1,1\}$.

Keywords: Harmonic numbers, $p$-adic valuation, asymptotic density, recurring sequences

MSC numbers: 11B75, 11B83

Supported by: The authors are supported by the National Natural Science Foundation of China, No.11771211 and NUPTSF, Grant No.NY220092.