iframe-proxy | Sunbelt Computer Software

46 lines (34 loc) · 1.27 KB

"""

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: 3.5 sec

"""

def solution(limit: int = 1_000_000) -> int:

"""

Returns an integer, the solution to the problem

>>> solution(10)

>>> solution(100)

3043

>>> solution(1_000)

304191

"""

phi = [i - 1 for i in range(limit + 1)]

for i in range(2, limit + 1):

for j in range(2 * i, limit + 1, i):

phi[j] -= phi[i]

return sum(phi[2 : limit + 1])

if __name__ == "__main__":

print(solution())

Sunbelt Computer Software