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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?
"""
defsolution(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)
forpinrange(3, limit, 2):
ifpnotinprimes:
continue
primes.difference_update(set(range(p*p, limit, p)))
phi= [float(n) forninrange(limit+1)]
forpinprimes:
forninrange(p, limit+1, p):
phi[n] *=1-1/p
returnint(sum(phi[2:]))
if__name__=="__main__":
print(f"{solution() =}")