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problem21.py
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problem21.py
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#!/usr/bin/env python
# -*- coding: utf-8 -*-
"""
Project Euler Problem 21:
Let d(n) be defined as the sum of proper divisors of n (numbers less than n which divide evenly into n).
If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and each of a and b are called amicable
numbers.
For example, the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110; therefore d(220) = 284. The
proper divisors of 284 are 1, 2, 4, 71 and 142; so d(284) = 220.
Evaluate the sum of all the amicable numbers under 10000.
"""
import math
memoized_results = {1: 1}
def sum_of_proper_divisors(n):
if n in memoized_results:
return memoized_results[n]
result = 1
sqrt = math.sqrt(n)
for divisor in range(2, int(sqrt)): # perfect squares will be dealt with later
other_divisor, remainder = divmod(n, divisor)
if remainder == 0:
result += divisor
result += other_divisor
if sqrt.is_integer():
# perfect square: the square root is also a proper divisor
result += int(sqrt)
memoized_results[n] = result
return result
amicable_sum = 0
for a in range(2, 10000):
b = sum_of_proper_divisors(a)
if a != b and sum_of_proper_divisors(b) == a:
amicable_sum += a
print(amicable_sum)