## Question

Consider the fraction, $\frac{n}{d}$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\mathrm{HCF}(n,d)=1$, it is called a reduced proper fraction.

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

$\displaystyle \frac{1}{8}, \frac{1}{7}, \frac{1}{6}, \frac{1}{5}, \frac{1}{4}, \frac{2}{7}, \frac{1}{3}, \frac{3}{8}, \frac{2}{5}, \frac{3}{7}, \frac{1}{2}, \frac{4}{7}, \frac{3}{5}, \frac{5}{8}, \frac{2}{3}, \frac{5}{7}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \frac{7}{8}$

It can be seen that there are 21 elements in this set.

How many elements would be contained in the set of reduced proper fractions for $d \leq 1,000,000$?

farey :: Int -> Int
f n = (n*(n + 3)) quot 2 - sum [farey (n quot k) | k <- [2..n]]
main = print $farey 1000000 - farey 1 $ ghc -O2 -o counting-fractions counting-fractions.hs
$time ./counting-fractions real 0m0.398s user 0m0.356s sys 0m0.036s ## Python #!/usr/bin/env python import math from operator import mul from functools import reduce def prime_factors(n): res = set() # iterate over all even numbers first. while n % 2 == 0: res.add(2) n //= 2 # try odd numbers up to sqrt(n) limit = math.sqrt(n+1) i = 3 while i <= limit: if n % i == 0: res.add(i) n //= i limit = math.sqrt(n+i) else: i += 2 if n != 1: res.add(n) return res def totient(n): if n == 1: return 1 return int(round(n * reduce(mul, [1 - 1.0 / p for p in prime_factors(n)]))) def farey_length(n): return sum(totient(m) for m in range(1, n+1)) - 1 def main(): print(farey_length(1000000)) if __name__ == "__main__": main() $ time python3 farey.py
sys    0m0.012s