57 lines
2.0 KiB
Python
57 lines
2.0 KiB
Python
#!/usr/bin/env python3
|
|
|
|
# A perfect number is a number for which the sum of its proper divisors is exactly equal to the number.
|
|
# For example, the sum of the proper divisors of 28 would be 1 + 2 + 4 + 7 + 14 = 28, which means that 28 is a perfect number.
|
|
#
|
|
# A number n is called deficient if the sum of its proper divisors is less than n and it is called abundant if this sum exceeds n.
|
|
#
|
|
# As 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the smallest number that can be written as the sum of two abundant numbers is 24.
|
|
# By mathematical analysis, it can be shown that all integers greater than 28123 can be written as the sum of two abundant numbers.
|
|
# However, this upper limit cannot be reduced any further by analysis even though it is known that the greatest number that cannot be expressed
|
|
# as the sum of two abundant numbers is less than this limit.
|
|
#
|
|
# Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.
|
|
|
|
from projecteuler import sum_of_divisors, timing
|
|
|
|
|
|
def is_abundant(n: int) -> bool:
|
|
return sum_of_divisors(n) > n
|
|
|
|
|
|
@timing
|
|
def p023() -> None:
|
|
ab_nums = [False] * 28124
|
|
|
|
# Find all abundant numbers smaller than 28123.
|
|
for i in range(12, 28124):
|
|
ab_nums[i] = is_abundant(i)
|
|
|
|
sums = [False] * 28124
|
|
|
|
# For every abundant number, sum every other abundant number greater
|
|
# than itself, until the sum exceeds 28123. Record that the resulting
|
|
# number is the sum of two abundant numbers.
|
|
for i in range(1, 28123):
|
|
if ab_nums[i]:
|
|
for j in range(i, 28123):
|
|
if ab_nums[j]:
|
|
_sum = i + j
|
|
|
|
if _sum <= 28123:
|
|
sums[_sum] = True
|
|
|
|
_sum = 0
|
|
|
|
# Sum every number that was not found as a sum of two abundant numbers.
|
|
for i in range(1, 28124):
|
|
if not sums[i]:
|
|
_sum = _sum + i
|
|
|
|
print('Project Euler, Problem 23')
|
|
print(f'Answer: {_sum}')
|
|
|
|
|
|
if __name__ == '__main__':
|
|
p023()
|