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53 lines
1.6 KiB
Python
53 lines
1.6 KiB
Python
"""
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A perfect number is a number for which the sum of its proper divisors is exactly
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equal to the number. For example, the sum of the proper divisors of 28 would be
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1 + 2 + 4 + 7 + 14 = 28, which means that 28 is a perfect number.
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A number n is called deficient if the sum of its proper divisors is less than n
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and it is called abundant if this sum exceeds n.
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As 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the smallest
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number that can be written as the sum of two abundant numbers is 24. By
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mathematical analysis, it can be shown that all integers greater than 28123
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can be written as the sum of two abundant numbers. However, this upper limit
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cannot be reduced any further by analysis even though it is known that the
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greatest number that cannot be expressed as the sum of two abundant numbers
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is less than this limit.
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Find the sum of all the positive integers which cannot be written as the sum
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of two abundant numbers.
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"""
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def solution(limit=28123):
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"""
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Finds the sum of all the positive integers which cannot be written as
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the sum of two abundant numbers
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as described by the statement above.
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>>> solution()
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4179871
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"""
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sumDivs = [1] * (limit + 1)
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for i in range(2, int(limit**0.5) + 1):
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sumDivs[i * i] += i
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for k in range(i + 1, limit // i + 1):
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sumDivs[k * i] += k + i
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abundants = set()
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res = 0
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for n in range(1, limit + 1):
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if sumDivs[n] > n:
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abundants.add(n)
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if not any((n - a in abundants) for a in abundants):
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res += n
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return res
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if __name__ == "__main__":
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print(solution())
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