Least simultaneous quadratic non-residue The only fly in the ointment is if $n_p=q$ or $n_q=p$. But the OP is perhaps not interested in these pathological cases.

A remarkable sum over partitions If you multiply both sides by $n!$ the LHS is the total number of permutations in $S_n$ and the RHS counts those with a given cycle structure $k_1$ cycles of length $a_1$, etc.