For a strictly increasing sequence \(A=\{a_i\}_{i=1}^{\infty }\) of positive integers, Borwein (Can Math Bull 20:117–118, 1978) proved that \(\sum _{i=1}^{n} { {{\,\mathrm{lcm}\,}}(a_i,a_{i+1})}^{-1}\le 1-\frac{1}{2^n}\) for any positive integer n, where the equality holds if and only if \(a_i=2^{i-1}\) for \(i=1,2, \ldots ,n+1\). Let r be an integer with \(3\le r\le 7\), Qian (C R Acad Sci Paris Ser I 355:1127–1132, 2017) further proved that \(\sum _{i=1}^{n} { {{\,\mathrm{lcm}\,}}(a_i, \ldots ,a_{i+r-1})}^{-1}\le U_r(n)\) and characterized the equality, where \(U_r(n)\) depends only on r and n. In this paper, under the condition \( {{\,\mathrm{lcm}\,}}(a_1, \ldots ,a_{r-1})\le a_r\), we determine the best upper bound (uniformly dependent only on r and n) of \(\sum _{i=1}^{n} { {{\,\mathrm{lcm}\,}}(a_i, \ldots ,a_{i+r-1})}^{-1}\) and also characterize the terms \(a_1,a_2, \ldots ,a_{n+r-1}\) such that the best upper bound is attained.

对于正整数的严格递增序列\(A=\{a_i\}_{i=1}^{\infty }\)，Borwein (Can Math Bull 20:117–118, 1978) 证明了\(\sum _{i=1}^{n} { {{\,\mathrm{lcm}\,}}(a_i,a_{i+1})}^{-1}\le 1-\frac{1}{ 2^n}\)对于任何正整数n，其中等式成立当且仅当\(a_i=2^{i-1}\)对于\(i=1,2, \ldots ,n+1\) . 令r为整数，其中\(3\le r\le 7\)，Qian (CR Acad Sci Paris Ser I 355:1127–1132, 2017) 进一步证明了\(\sum _{i=1}^{n } { {{\,\mathrm{lcm}\,}}(a_i, \ldots ,a_{i+r-1})}^{-1}\le U_r(n)\)并表征了等式，其中\(U_r(n)\)仅取决于r和n。在本文中，在条件\( {{\,\mathrm{lcm}\,}}(a_1, \ldots ,a_{r-1})\le a_r\) 下，我们确定了最佳上限（一致依赖仅在r和n ) 的\(\sum _{i=1}^{n} { {{\,\mathrm{lcm}\,}}(a_i, \ldots ,a_{i+r-1}) }^{-1}\)并表征项\(a_1,a_2, \ldots ,a_{n+r-1}\)以便获得最佳上限。