On the denominators of $\varepsilon$-harmonic numbers
Bull. Korean Math. Soc. 2020 Vol. 57, No. 6, 1383-1392
https://doi.org/10.4134/BKMS.b191047
Published online September 1, 2020
Printed November 30, 2020
Bing-Ling Wu, Xiao-Hui Yan
Nanjing University of Posts and Telecommunications; Anhui Normal University
Abstract : 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.
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