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All Random Solved Random Open
OPEN
Let $A=\{a_1<a_2<\cdots\}\subseteq \mathbb{N}$ be infinite and let $A(x)$ count the numer of indices for which $\mathrm{lcm}(a_i,a_{i+1})\leq x$. Is it true that $A(x) \ll x^{1/2}$? How large can \[\liminf \frac{A(x)}{x^{1/2}}\] be?
It is easy to give a sequence with \[\limsup\frac{A(x)}{x^{1/2}}=c>0.\] There are related results (particularly for the more general case of $\mathrm{lcm}(a_i,a_{i+1},\ldots,a_{i+k})$) in a paper of Erdős and Szemerédi [ErSz80].
Additional thanks to: Zachary Chase