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51 lines
1.4 KiB
Python
51 lines
1.4 KiB
Python
"""
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Problem 72 Counting fractions: https://projecteuler.net/problem=72
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Description:
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Consider the fraction, n/d, where n and d are positive integers. If n<d and HCF(n,d)=1,
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it is called a reduced proper fraction.
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If we list the set of reduced proper fractions for d ≤ 8 in ascending order of size, we
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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,
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3/4, 4/5, 5/6, 6/7, 7/8
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It can be seen that there are 21 elements in this set.
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How many elements would be contained in the set of reduced proper fractions for
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d ≤ 1,000,000?
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Solution:
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Number of numbers between 1 and n that are coprime to n is given by the Euler's Totient
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function, phi(n). So, the answer is simply the sum of phi(n) for 2 <= n <= 1,000,000
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Sum of phi(d), for all d|n = n. This result can be used to find phi(n) using a sieve.
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Time: 1 sec
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"""
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import numpy as np
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def solution(limit: int = 1_000_000) -> int:
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"""
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Returns an integer, the solution to the problem
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>>> solution(10)
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31
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>>> solution(100)
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3043
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>>> solution(1_000)
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304191
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"""
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# generating an array from -1 to limit
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phi = np.arange(-1, limit)
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for i in range(2, limit + 1):
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if phi[i] == i - 1:
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ind = np.arange(2 * i, limit + 1, i) # indexes for selection
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phi[ind] -= phi[ind] // i
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return int(np.sum(phi[2 : limit + 1]))
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if __name__ == "__main__":
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print(solution())
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