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59 lines
1.5 KiB
Python
59 lines
1.5 KiB
Python
"""
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Project Euler Problem 187: https://projecteuler.net/problem=187
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A composite is a number containing at least two prime factors.
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For example, 15 = 3 x 5; 9 = 3 x 3; 12 = 2 x 2 x 3.
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There are ten composites below thirty containing precisely two,
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not necessarily distinct, prime factors: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26.
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How many composite integers, n < 10^8, have precisely two,
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not necessarily distinct, prime factors?
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"""
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from math import isqrt
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def calculate_prime_numbers(max_number: int) -> list[int]:
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"""
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Returns prime numbers below max_number
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>>> calculate_prime_numbers(10)
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[2, 3, 5, 7]
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"""
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is_prime = [True] * max_number
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for i in range(2, isqrt(max_number - 1) + 1):
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if is_prime[i]:
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for j in range(i**2, max_number, i):
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is_prime[j] = False
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return [i for i in range(2, max_number) if is_prime[i]]
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def solution(max_number: int = 10**8) -> int:
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"""
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Returns the number of composite integers below max_number have precisely two,
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not necessarily distinct, prime factors
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>>> solution(30)
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10
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"""
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prime_numbers = calculate_prime_numbers(max_number // 2)
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semiprimes_count = 0
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left = 0
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right = len(prime_numbers) - 1
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while left <= right:
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while prime_numbers[left] * prime_numbers[right] >= max_number:
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right -= 1
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semiprimes_count += right - left + 1
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left += 1
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return semiprimes_count
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if __name__ == "__main__":
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print(f"{solution() = }")
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