首页|The exact upper bound for the sum of reciprocals of least common multiples
The exact upper bound for the sum of reciprocals of least common multiples
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NSTL
Elsevier
Let r be a positive number with r >= 2 and let A = {a(i)}(i=1)(infinity), be an arbitrarily given strictly increasing sequence of positive integers. Let S-r(A).:= Sigma(infinity)(i=1)1/[a(i) + a(i+1), ... ,a(i+r-1)]. In 1978, Borwein obtained S-2(A) <= 1 with equality occurring if and only if a(i) = 2(i-1) for i >= 1. Qian and Zhao et al. obtained exact upper bounds for S-r(A) as 3 <= r <= 7 and 8 <= r <= 11 respectively in 2017 and 2019. In this paper, we give several methods to obtain the upper bounds for S-12(A) and S-13(A), with explicit sequences which reach the corresponding upper bounds. We propose a conjecture that the exact upper bounds for S-r(A) are tau(h)-r+2/h for all r >= 2, where h is a highly composite number and tau(h) denotes the number of divisors of h. In addition, we offer some sequences that reach the exact upper bounds. (C) 2021 Elsevier Inc. All rights reserved.
Least common multipleReciprocalExact upper boundHighly composite numberDivisor sequencer-optimal sequenceNONTRIVIAL LOWER BOUNDSIMPROVEMENTSSEQUENCES
NSTL
Elsevier
