I'm curious about the following miscellaneous conjectures, for which I hope that one can to get a counterexample.
I add the encyclopedia Wikipedia's article for the Airy functions $\operatorname{Ai}(x)$ and $\operatorname{Bi}(x)$, and we denote the floor function as $\lfloor x\rfloor$. Then playing with Wolfram Alpha online calculator I've defined two miscellaneous sequences.
Definition. For integers $n\geq 1$ we define the sequences of integers $$a_n=-4+\biggl\lfloor\frac{1}{\operatorname{Ai}(n)-\operatorname{Ai}(n+1)}\biggr\rfloor,$$ and $$b_n=-2+\biggl\lfloor \operatorname{Bi}(n+1)-\operatorname{Bi}(n)\biggr\rfloor.$$
(From few computational evidence, the calculations that I got) I've consider the following conjecture.
Conjectures. If $a_n\equiv 1\text{ mod }2$, then this odd integer $a_n$ has no repeated prime factors. And similarly, if $b_n\equiv 1\text{ mod }2$, then this odd integer $b_n$ has no repeated prime factors.
Question. I'm curious to know what is the smallest counterexample for each of previous miscellaneous conjectures. I think that it should be possible to find a counterexample, any case feel free to add if there is some reasoning or heuristic to find the counterexamples that I evoke. Many thanks and good day.