# Amicable numbers Project Euler 第 21 题

## Question

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.

## Analysis

1. 因子求和函数应该怎样优化？
2. 字典来保存计算结果。
3. 官方思路

## Program

``````from math import sqrt, ceil

def get_sum(num):
sum = 1
mid = sqrt(num)
for factor in range(2, ceil(mid) + 1):
if num % factor == 0:
sum += factor
if factor != mid:
sum += num // factor

return sum

max_size = 10000
result = 0
cal = {}

for i in range(1, max_size):
if cal.__contains__(i):
a = cal[i]
else:
a = get_sum(i)
cal[i] = a

if cal.__contains__(a):
b = cal[a]
else:
b = get_sum(a)
cal[a] = b

if i == b and i != a:
result += i

print(result)
``````