Question 118
For any integer $n \gt 1$, let $s(n)$ be its smallest prime divisor and $d(n)$ be the number of its positive divisors. Is it possible to choose $2022$ positive integers $a_1, a_2, \dots, a_{2022}$ with $a_1 \lt a_2 - 1 \lt \dots \lt a_{2022} - 2021$ such that for all $k = 1, \dots, 2021$ it holds that
$$d(a_{k+1} - a_k - 1) \gt 2022^k \text{ and } s(a_{k+1} - a_k) \gt 2022^k?$$
$$d(a_{k+1} - a_k - 1) \gt 2022^k \text{ and } s(a_{k+1} - a_k) \gt 2022^k?$$