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Problem 73

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Counting Fractions in a Range

Consider the fraction, $n/d$, where $n$ and $d$ are positive integers. If $n<d$ and $HCF(n,d)=1$, it is called a reduced proper fraction.

If we list the set of reduced proper fractions for $d\le 8$ in ascending order of size, we get:

$$1/8,1/7,1/6,1/5,1/4,2/7,1/3,\mathbf{\text{3/8, 2/5, 3/7}},1/2,$$
$$4/7,3/5,5/8,2/3,5/7,3/4,4/5,5/6,6/7,7/8$$

It can be seen that there are $3$ fractions between $1/3$ and $1/2$.

How many fractions lie between $1/3$ and $1/2$ in the sorted set of reduced proper fractions for $d\le 12,000$?

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分数有范围计数

考虑形如$n/d$的分数，其中$n$和$d$均为正整数。如果$n<d$且其最大公约数为$1$，则称该分数为最简真分数。

将所有$d\le 8$的最简真分数构成的集合按大小升序排列：

$$1/8,1/7,1/6,1/5,1/4,2/7,1/3,\mathbf{\text{3/8, 2/5, 3/7}},1/2,$$
$$4/7,3/5,5/8,2/3,5/7,3/4,4/5,5/6,6/7,7/8$$

可以看出在$1/3$和$1/2$之间有$3$个分数。

将$d\le 12,000$的最简真分数构成的集合排序后，在$1/3$和$1/2$之间有多少个分数？

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