overengineering

from primefac import primefac
from itertools import combinations
from operator import mul


def sum_divisible_by_p(p, start, limit):
    stop = int((limit-1)/float(p))
    return p*(stop+start)*(stop-start+1)/2


def sum_divisible_fast(divisors, start, stop):
    proper_divisors = []
    for div in divisors:
        proper_divisors.extend(list(primefac(div)))
    divisors = list(set(proper_divisors))

    total = 0
    for divisor in divisors:
        total += sum_divisible_by_p(divisor, start, stop)

    for i in range(2, len(divisors)+1):
        k = (-1)**(i-1)
        for perm in combinations(divisors, i):
            product = reduce(mul, list(perm))
            total += k*sum_divisible_by_p(product, start, stop)
    return int(total)


def sum_divisible(divisors=[3, 5], start=0, stop=100):
    count = 0
    for num in xrange(start, stop):
        for d in divisors:
            if num % d == 0:
                count += num
                break
    return count

if __name__ == "__main__":

    print sum_divisible([3, 5, 7, 13], 1, 100000)
    print sum_divisible_fast([3, 5, 7, 13], 1, 100000)