Problem 73

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:

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

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 $d \le 12,000$?

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$d\le8$的最简真分数构成的集合按大小升序排列：

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

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？