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On the sum of reciprocals of least common multiples, II

Author

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  • Zhao, Jianrong
  • Qian, Guoyou
  • Zhao, Wei

Abstract

Let n and r be positive integers with r≥2 and let A={ai}i=1∞ be a strictly increasing sequence of positive integers. Let SA,r(n):=∑i=1n1lcm(ai,…,ai+r−1). In 1978, Borwein showed that SA,2(n)≤1−12n with the equality occurring if and only if ai=2i−1 for 1≤i≤n+1. In 2017, Qian proved that SA,r(n)≤Vr(n) for 3≤r≤7 and characterized the first n+r−1 terms of the sequence A such that SA,r(n)=Vr(n) holds, where Vr(n) depends only on r and n. In this paper, we further investigate SA,r(n) for 8≤r≤11 and we obtain the least upper bound Ur of SA,r(n) for all strictly increasing sequences A of positive integers and for all positive integers n, where Ur is a constant depending only on r.

Suggested Citation

  • Zhao, Jianrong & Qian, Guoyou & Zhao, Wei, 2021. "On the sum of reciprocals of least common multiples, II," Applied Mathematics and Computation, Elsevier, vol. 399(C).
    • Handle: RePEc:eee:apmaco:v:399:y:2021:i:c:s0096300321000515
    DOI: 10.1016/j.amc.2021.126003
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    Cited by:

    1. Chiu, Sunben & He, Yuqing & Yuan, Pingzhi, 2022. "The exact upper bound for the sum of reciprocals of least common multiples," Applied Mathematics and Computation, Elsevier, vol. 416(C).

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