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


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