BALKAN-SL 2018 N1

For positive integers $m$ and $n$, let $d(m, n)$ be the number of distinct primes that divide both $m$ and $n$. For instance, $d(60,126)=d\left(2^{2} \times 3 \times 5,2 \times 3^{2} \times 7\right)=2$. Does there exist a sequence $\left(a_{n}\right)$ of positive integers such that: (i) $a_{1} \geqslant 2018^{2018}$; (ii) $a_{m} \leqslant a_{n}$ whenever $m \leqslant n$; (iii) $d(m, n)=d\left(a_{m}, a_{n}\right)$ for all positive integers $m \neq n$ ? (United Kingdom)