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  • 1 School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, P. R. China
  • 2 School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, P. R. China and School of Mathematics and Statistics, Anhui Normal University, Wuhu 241003, P. R. China
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Abstract

Let Hn be the n-th harmonic number and let vn be its denominator. It is known that vn is even for every integer n>=2. In this paper, we study the properties of Hn and prove that for any integer n, vn = en(1+o(1)). In addition, we obtain some results of the logarithmic density of harmonic numbers.

  • [1]

    Boyd, D. W., A p-adic study of the partial sums of the harmonic series, Exp. Math., 3(4 ) (1994), 287302.

  • [2]

    Duncan, R. L., Note on the initial digit problem, Fibonacci Quart ., 7(5 ) (1969), 474475.

  • [3]

    Eswarathasan, A. and Levine, E., p-integral harmonic sums, Discrete Math ., 91(3 ) (1991), 249257.B.-L. WU and X.-H. YAN: Some Properties of Harmonic Numbers.

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  • [4]

    LEONETTI, P. and Sanna, C., On the p-adic valuation of Stirling numbers of the first kind, Acta Math. Hungar., 151 (2017), 217231.

  • [5]

    SANNA, C., On the p-adic valuation of harmonic numbers, J. Number Theory, 166 (2016), 4146.

  • [6]

    SHIU, P., The denominators of harmonic numbers, arXiv:1607.02863v1.

  • [7]

    Wu, B. L. and Chen, Y. G., On certain properties of harmonic numbers, J. Number Theory, 175 (2017), 6686.

  • [8]

    Wu, B. L. and Chen, Y. G., On the denominators of harmonic numbers, C. R. Acad. Sci. Paris, Ser I 356 (2018), 129132.

  • [9]

    Wu, B. L. and Chen, Y. G., On the denominators of harmonic numbers, II, J. Number Theory, 200 (2019), 397406.