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Problem 889

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Rational Blancmange

Recall the blancmange function from Problem 226: $T(x)=\sum _{n=0}^{\mathrm{\infty }}{\displaystyle \frac{s(2^{n}x)}{2^n}}$, where $s(x)$ is the distance from $x$ to the nearest integer.

For positive integers $k,t,r$, we write
$$F(k,t,r)=(2^{2k}-1)T\left(\frac{(2^t+1{)}^r}{2^k+1}\right).$$
It can be shown that $F(k,t,r)$ is always an integer.

For example, $F(3,1,1)=42$, $F(13,3,3)=23093880$ and $F(103,13,6)\equiv 878922518(\mathrm{mod}1000062031)$.

Find $F({10}^{18}+31,{10}^{14}+31,62)$. Give your answer modulo $1000062031$.

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有理牛奶冻

考虑226题中提及的牛奶冻函数：$T(x)=\sum _{n=0}^{\mathrm{\infty }}{\displaystyle \frac{s(2^{n}x)}{2^n}}$，其中$s(x)$是$x$到最近整数的距离。

对于正整数$k,t,r$，记：
$$F(k,t,r)=(2^{2k}-1)T\left(\frac{(2^t+1{)}^r}{2^k+1}\right)$$
可以证明$F(k,t,r)$总是一个整数。

例如，$F(3,1,1)=42$，$F(13,3,3)=23093880$，$F(103,13,6)\equiv 878922518(\mathrm{mod}1000062031)$。

求$F({10}^{18}+31,{10}^{14}+31,62)$，并对$1000062031$取余作为你的答案。

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