## Conjecture concerning sequences with a lot of consecutive elements with small lcm

Let $1 \le a_1 < a_2 < ..$ be an infinite sequence of positive integers, such that, if we define $A(n)$ to be the number of indices $i$ such that $\text{lcm}(a_i, a_{i+1})\le n$, then we have for every $\epsilon > 0$ infinitely many $n$ such that $A(n) > (c - \epsilon)\sqrt{n}$, where $c = \displaystyle \sum_{j=1}^{\infty} \dfrac{1}{(j+1)\sqrt{j}} \approx 1.86$. In my previous post I claimed that such sequences exist, and in due time I will prove this claim, but for now, let’s assume it.

Now, let $\epsilon$ be small and $n_1 < n_2 < ..$ be the sequence of all $n$ such that $A(n) > (c - \epsilon)\sqrt{n}$. Then I’d like to bluntly conjecture that there exist infinitely many $i$ such that $\dfrac{n_{i+1}}{n_i} > c_1\epsilon^{-4}$, for some absolute $c_1$. In particular, $\displaystyle \limsup \dfrac{n_{i+1}}{n_i}$ goes to $\infty$ as $\epsilon$ goes to $0$, that is: $\displaystyle \lim_{\epsilon \rightarrow 0} \limsup_{i \rightarrow \infty} \dfrac{n_{i+1}}{n_i} = \infty$