Project Euler

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.

def proper_divisors(n):
  """
  Find all divisors of n by trial division
  """
  divisors = set([1])
  for i in range(2, math.ceil(n ** 0.5)+1):
      if n % i == 0:
        divisors.add(i)
        divisors.add(n/i)
  return divisors
 
def amicable_numbers(n):
  candidates = range(1,n)
  amicables = []
  for a in candidates:
    b = sum(proper_divisors(a))
    if a != b \
        and sum(proper_divisors(a)) == b \
        and sum(proper_divisors(b)) == a:
      amicables.append(a)
      amicables.append(b)
      candidates.remove(b)
  return amicables
 
>>> sum(amicable_numbers(10000))
31626

Tags: amicable, divisors, euler, factors, problem21, python

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