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Problem 889

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Rational Blancmange

Recall the blancmange function from Problem 226: $T(x) = \sum\limits_{n = 0}^\infty\dfrac{s(2^nx)}{2^n}$, 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 $F(103, 13, 6) \equiv 878922518\pmod {1\ 000\ 062\ 031}$.

Find $F(10^{18} + 31, 10^{14} + 31, 62)$. Give your answer modulo $1\ 000\ 062\ 031$.

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有理牛奶冻

考虑226题中提及的牛奶冻函数：$T(x) = \sum\limits_{n = 0}^\infty\dfrac{s(2^nx)}{2^n}$，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，$F(3, 1, 1) = 42$，$F(13, 3, 3) = 23093880$，$F(103, 13, 6) \equiv 878922518\pmod {1\ 000\ 062\ 031}$。

求$F(10^{18} + 31, 10^{14} + 31, 62)$，并对$1\ 000\ 062\ 031$取余作为你的答案。

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