= Consecutive integers divisible by prescribed prime powers
{title2=$q_i^{a_i}\mid n+i\quad(1\leq i\leq r)$}
Let $q_1,\ldots,q_r$ be distinct <prime numbers> and $a_1,\ldots,a_r$ 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 $n\equiv-i\pmod{q_i^{a_i}}$ 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>.
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