Define $Q(n)$ to be the smallest number that occurs in exactly $n$ Pythagorean triples $(a,b,c)$ where $a<b<c$.

For example, $15$ is the smallest number occurring in exactly $5$ Pythagorean triples: $$(9,12,\mathbf{15})(8,\mathbf{15},17)(\mathbf{15},20,25)(\mathbf{15},36,39)(\mathbf{15},112,113)$$ and so $Q(5)=15$.

You are also given $Q(10)=48$ and $Q({10}^3)=8064000$.

Find ${\displaystyle \sum _{k=1}^{18}Q({10}^k)}$. Give your answer modulo $409120391$.