Pythagorean Triple Occurrence (noch nicht übersetzt)

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

For example, 15 is the smallest number occurring in exactly 5 Pythagorean triples: $$(9,12,\mathbf{15})\quad (8,\mathbf{15},17)\quad (\mathbf{15},20,25)\quad (\mathbf{15},36,39)\quad (\mathbf{15},112,113)$$ 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 $409120391$.