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Problem 946

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Continued Fraction Fraction

Given the representation of a continued fraction
$$ a_0+ \cfrac{1}{a_1+ \cfrac{1}{a_2+\cfrac{1}{a_3+\ddots}}}= [a_0;a_1,a_2,a_3,\ldots] $$

$\alpha$ is a real number with continued fraction representation:
$\alpha = [2;1,1,2,1,1,1,2,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,2,\ldots]$
where the number of $1$’s between each of the $2$’s are consecutive prime numbers.

$\beta$ is another real number defined as
$$\beta = \frac{2\alpha+3}{3\alpha+2}$$

The first ten coefficients of the continued fraction of $\beta$ are $[0;1,5,6,16,9,1,10,16,11]$ with sum $75$.

Find the sum of the first $10^8$ coefficients of the continued fraction of $\beta$.

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连分数的分数

任意连分数可以表示为
$$ a_0+ \cfrac{1}{a_1+ \cfrac{1}{a_2+\cfrac{1}{a_3+\ddots}}}= [a_0;a_1,a_2,a_3,\ldots] $$

$\alpha$是一个实数，其连分数表示为：
$\alpha = [2;1,1,2,1,1,1,2,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,2,\ldots]$
其中每两个$2$之间的$1$的数目是连续的质数。

$\beta$是另一个实数，定义为：
$$\beta = \frac{2\alpha+3}{3\alpha+2}$$

$\beta$的连分数表示的前十个系数是$[0;1,5,6,16,9,1,10,16,11]$，其和为$75$。

求$\beta$的连分数表示的前$10^8$个系数之和。

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