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Problem 827

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Pythagorean Triple Occurrence

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 $409120391$.

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毕达哥拉斯三元组频数

记$Q(n)$为最小的、恰好出现在$n$个毕达哥拉斯三元组$(a,b,c)$中的数，其中$a<b<c$。

例如，$15$是最小的、恰好出现在$5$个毕达哥拉斯三元组中的数：
$$(9,12,\mathbf{15})(8,\mathbf{15},17)(\mathbf{15},20,25)(\mathbf{15},36,39)(\mathbf{15},112,113)$$
因此$Q(5)=15$。

已知$Q(10)=48$，$Q({10}^3)=8064000$。

求${\displaystyle \sum _{k=1}^{18}Q({10}^k)}$，并将你的答案对$409120391$取余。

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