Consecutive integers divisible by prescribed prime powers ($q_i^{a_i}\mid n+i(1\le i\le r)$)

Let $q_1,\dots ,q_r$ be distinct prime numbers and $a_1,\dots ,a_r$ positive integers. There are infinitely many positive integers $n$ with $q_i^{a_i}\mid n+i$ for every $i$. Indeed the moduli $q_i^{a_i}$ are pairwise coprime, so the Chinese remainder theorem solves $n\equiv -i(\mathrm{mod}{q}_i^{a_i})$ simultaneously. Adding multiples of their product gives arbitrarily large positive solutions. This allows long runs in which every integer has a large prescribed prime power divisor, even though neighboring integers are coprime.