Let $p_k$ be the $k$th prime and, for convenience, $p_0=1.$
For any non-negative $i$, the $m$ positive numbers $p_i,p_{i+1},p_{i+2},\cdots , p_{i+m-1}$ are pairwise relatively prime. Try $p_i$ the smallest prime with $n$ bits (or $1$ for $n=1$). If that fails then you should be able to do it for $n+1$ bits. That might be as good as it gets.