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Problem 809

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Rational Recurrence Relation

The following is a function defined for all positive rational values of $x$.
$$ f(x)=\begin{cases} x & x\text{ is integral}\\
f(\frac 1{1-x}) & x \lt 1\\
f\Big(\frac 1{\lceil x\rceil -x}-1+f(x-1)\Big) & \text{otherwise}\end{cases} $$

For example, $f(3/2)=3$, $f(1/6) = 65533$ and $f(13/10) = 7625597484985$.

Find $f(22/7)$. Give your answer modulo $10^{15}$.

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有理数递归关系式

如下是定义在所有正有理数$x$上的函数：
$$ f(x)=\begin{cases} x & \text{若}x\text{是整数}\\
f(\frac 1{1-x}) & \text{若}x \lt 1\\
f\Big(\frac 1{\lceil x\rceil -x}-1+f(x-1)\Big) & \text{其余情况}\end{cases} $$

例如，$f(3/2)=3$，$f(1/6) = 65533$，$f(13/10) = 7625597484985$。

求$f(22/7)$，并将你的答案对$10^{15}$取余。

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