def product(numbers):
    r = 1
    for n in numbers:
        r = r * n
    return r

def _incrementer():
    """ Generator function that returns natural numbers where the higher
    digits are smaller or equal than the lower digits. """
    n_digits = 2
    r = [1] * n_digits
    while True:
        yield r
        incremented = False
        for i in range(n_digits - 1):
            if r[i] < r[i + 1]:
                r[i] += 1
                incremented = True
                for j in range(i):
                    r[j] = 1
                break
        if incremented:
            continue
        elif r[-1] < 2000:
            r[-1] += 1
            for j in range(n_digits - 1):
                r[j] = 1
        else:
            n_digits += 1
            r = [1] * n_digits

for k in range(40, 50):
    s = product_sum_number(k)
        print("k={}: {} = {}".format(k, product(s), s))


def product_sum(k):
    # Create initial list of canditates
    canditates = []
    for n in range(2, k + 1):
        rest_sum = k - n
        if rest_sum + n * 2 < 2 ** n:
            break
        canditates.append(tuple([2 for _ in range(n)]))
    processed = set(canditates)
    solutions = []

    while canditates:
        numbers = canditates.pop()
        processed.add(numbers)

        s = k - len(numbers) + sum(numbers)
        p = product(numbers)
        if s == p:
            solutions.append(numbers)
        elif s > p:
            for i in range(len(numbers) - 1):
                if numbers[i] < numbers[i + 1]:
                    n = list(numbers)
                    n[i] += 1
                    t = tuple(n)
                    if not t in processed:
                        canditates.append(t)

            n = list(numbers)
            n[-1] += 1
            t = tuple(n)
            if not t in processed:
                canditates.append(t)
        else:
            pass
    result = min([product(ns) for ns in solutions])
    return result


def euler_088():
    ns = set()
    for k in range(2, 12001):
        r = product_sum_2d(k)
        print("k={}: {}".format(k, r))
        ns.add(r)
    return sum(ns)


if __name__ == "__main__":
    solution = euler_088()
    print("e088.py: " + str(solution))
    assert(solution == 7587457)