标题： 关于某些序列中连续元素的互质性 显示英文标题

标题： On coprimality of consecutive elements in certain sequences

摘要： 研究在某些算术序列中寻找素数块是数论中的一个经典问题。 研究这种序列中连续元素的块，这些块中的元素两两互质，也是非常有趣的。 在此背景下，我们证明了如果$f$是$[1, \infty)$上的二次连续可微实值函数，并且当$x \to \infty$时$f''(x) \to 0$且$\limsup_{x \to \infty} f'(x) = \infty$，那么序列$(\lfloor f(n) \rfloor)_n$中存在任意长的连续两两互质元素块。 这个结果改进了第一作者、Drmota 和 Müllner 最近结果中的定性部分。 我们还证明存在一个子集$\mathcal{A} \subseteq \mathbb{N}$，其上Banach密度为1，使得对于任意两个不同的整数$m, n \in \mathcal{A}$，整数$\lfloor f(m) \rfloor$和$\lfloor f(n) \rfloor$都是两两互质的。 此外，我们证明在序列$(\lfloor f(n) \rfloor)_n$中存在任意长的连续元素块，其中没有两个元素是两两互质的。 显示英文摘要

摘要： The study of finding blocks of primes in certain arithmetic sequences is one of the classical problems in number theory. It is also very interesting to study blocks of consecutive elements in such sequences that are pairwise coprime. In this context, we show that if $f$ is a twice continuously differentiable real-valued function on $[1, \infty)$ such that $f''(x) \to 0$ as $x \to \infty$ and $\limsup_{x \to \infty} f'(x) = \infty$, then there exist arbitrarily long blocks of pairwise coprime consecutive elements in the sequence $(\lfloor f(n) \rfloor)_n$. This result refines the qualitative part of a recent result by the first author, Drmota and M\"{u}llner. We also prove that there exists a subset $\mathcal{A} \subseteq \mathbb{N}$ having upper Banach density one such that for any two distinct integers $m, n \in \mathcal{A}$, the integers $\lfloor f(m) \rfloor$ and $\lfloor f(n) \rfloor$ are pairwise coprime. Further, we show that there exist arbitrarily long blocks of consecutive elements in the sequence $(\lfloor f(n) \rfloor)_n$ such that no two of them are pairwise coprime.