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project_euler/problem_072/sol1.py
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""" Problem 72 Counting fractions: https://projecteuler.net/problem=72 Description: 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 ≤ 8 in ascending order of size, we get: 1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 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 21 elements in this set. How many elements would be contained in the set of reduced proper fractions for d ≤ 1,000,000? Solution: Number of numbers between 1 and n that are coprime to n is given by the Euler's Totient function, phi(n). So, the answer is simply the sum of phi(n) for 2 <= n <= 1,000,000 Sum of phi(d), for all d|n = n. This result can be used to find phi(n) using a sieve. Time: 1 sec """ import numpy as np def solution(limit: int = 1_000_000) -> int: """ Returns an integer, the solution to the problem >>> solution(10) 31 >>> solution(100) 3043 >>> solution(1_000) 304191 """ # generating an array from -1 to limit phi = np.arange(-1, limit) for i in range(2, limit + 1): if phi[i] == i - 1: ind = np.arange(2 * i, limit + 1, i) # indexes for selection phi[ind] -= phi[ind] // i return int(np.sum(phi[2 : limit + 1])) if __name__ == "__main__": print(solution())