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83 lines
2.0 KiB
Python
83 lines
2.0 KiB
Python
"""
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Project Euler Problem 77: https://projecteuler.net/problem=77
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It is possible to write ten as the sum of primes in exactly five different ways:
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7 + 3
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5 + 5
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5 + 3 + 2
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3 + 3 + 2 + 2
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2 + 2 + 2 + 2 + 2
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What is the first value which can be written as the sum of primes in over
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five thousand different ways?
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"""
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from __future__ import annotations
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from functools import lru_cache
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from math import ceil
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NUM_PRIMES = 100
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primes = set(range(3, NUM_PRIMES, 2))
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primes.add(2)
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prime: int
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for prime in range(3, ceil(NUM_PRIMES**0.5), 2):
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if prime not in primes:
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continue
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primes.difference_update(set(range(prime * prime, NUM_PRIMES, prime)))
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@lru_cache(maxsize=100)
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def partition(number_to_partition: int) -> set[int]:
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"""
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Return a set of integers corresponding to unique prime partitions of n.
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The unique prime partitions can be represented as unique prime decompositions,
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e.g. (7+3) <-> 7*3 = 12, (3+3+2+2) = 3*3*2*2 = 36
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>>> partition(10)
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{32, 36, 21, 25, 30}
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>>> partition(15)
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{192, 160, 105, 44, 112, 243, 180, 150, 216, 26, 125, 126}
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>>> len(partition(20))
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26
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"""
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if number_to_partition < 0:
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return set()
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elif number_to_partition == 0:
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return {1}
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ret: set[int] = set()
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prime: int
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sub: int
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for prime in primes:
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if prime > number_to_partition:
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continue
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for sub in partition(number_to_partition - prime):
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ret.add(sub * prime)
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return ret
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def solution(number_unique_partitions: int = 5000) -> int | None:
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"""
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Return the smallest integer that can be written as the sum of primes in over
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m unique ways.
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>>> solution(4)
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10
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>>> solution(500)
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45
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>>> solution(1000)
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53
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"""
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for number_to_partition in range(1, NUM_PRIMES):
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if len(partition(number_to_partition)) > number_unique_partitions:
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return number_to_partition
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return None
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if __name__ == "__main__":
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print(f"{solution() = }")
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