Python/project_euler/problem_072/sol2.py

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"""
Project Euler Problem 72: https://projecteuler.net/problem=72
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?
"""
def solution(limit: int = 1000000) -> int:
"""
Return the number of reduced proper fractions with denominator less than limit.
>>> solution(8)
21
>>> solution(1000)
304191
"""
primes = set(range(3, limit, 2))
primes.add(2)
for p in range(3, limit, 2):
if p not in primes:
continue
primes.difference_update(set(range(p * p, limit, p)))
phi = [float(n) for n in range(limit + 1)]
for p in primes:
for n in range(p, limit + 1, p):
phi[n] *= 1 - 1 / p
return int(sum(phi[2:]))
if __name__ == "__main__":
print(f"{solution() = }")