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It has been shown here that there exist positive integers $n,k$ such that for all primes $p$ and integers $q\ge 1$, $m\ge 2$, we have $$ pq^m \not\equiv k\bmod{n}. $$

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Hence, let $f(n)$ be the number of $k \in \{1,2,\ldots,n\}$ such that $pq^m \equiv k\bmod{n}$, for some prime $p$ and integers $q\ge 1$, $m\ge 2$. (In particular, $f(n) \le n-1$ for some $n$.)

Question. Is it true that there exists $c>0$ such that $f(n) \ge cn$ for all $n$?

Ps. Thanks to Erick Wong for the correction

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