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Let $S$ be the set of positive integers not divisible by $3$ where if $p$ is a prime factor of $n \in S$ and $p \equiv 1\bmod 3$ then $p^2$ does not divide $n$, but if $p\equiv2 \bmod 3$ then $p^2$ does divide $n$ but $p^3$ does not.
So $S$ is, in a way, square free with respect to primes congrue...
