mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-11-24 05:21:09 +00:00
46 lines
1.2 KiB
Python
46 lines
1.2 KiB
Python
|
"""
|
||
|
|
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() = }")
|