from collections import Counter
from functools import reduce

def sum_proper_divisors (nbr):
    proper_divisors = divisors(nbr)[0:-1]

    return reduce(lambda x, y: x + y, proper_divisors)

def divisors(num):
    result = []
    def recurse(tuples, product = 1):
        factor, count = tuples[0]

        for i in range(0, count + 1):
            if (i != 0):
                product = product * factor

            if (len(tuples) > 1):
                recurse(tuples[1:], product)
            else:
                result.append(product)

    factors = find_prime_factors(num)

    recurse(list(Counter(factors).items()))

    result.sort()

    return result


def find_largest_prime_factor(n):
    prime_factor = 1
    factors.append(prime_factor)

    print('appended')
    i = 2

    while i <= n / i:
        if n % i == 0:
            factors.append(i)
            prime_factor = i
            n = int(n / i)
        else:
            i += 1

    if prime_factor < n:
        prime_factor = n

    return prime_factor


def find_prime_factors(n):
    factors = []

    i = 2

    while i <= n / i:
        if n % i == 0:
            factors.append(i)
            n = int(n / i)
        else:
            i += 1

    factors.append(n)

    return factors

def factorial(n):
    return reduce(lambda x, y: x * y, range(1, n + 1))