from math import gcd
from functools import lru_cache


@lru_cache
def r(n):
    assert n > 0
    if n == 1:
        return 1
    else:
        return 10**(n - 1) + r(n - 1)


def r_closed_form(n):
    assert n > 0
    return (10**n - 1) // 9


def r_modulo_closed_form(n, m):
    assert n > 0 and m > 0
    return ((pow(10, n, 9 * m) - 1) // 9) % m


def a(n):
    assert gcd(n, 10) == 1
    k = 1
    while True:
        if r_modulo_closed_form(k, n) == 0:
            return k
        k += 1


def euler_129():

    # Comparing naiv to efficient implementations to find potential bugs.
    for n in range(5, 1000):
        assert r(n) == r_closed_form(n)
        for m in range(2, 20):
            assert (r_closed_form(n) % m)  == r_modulo_closed_form(n, m)

    assert a(7) == 6
    assert a(41) == 5
    assert a(17) == 16

    target = 10**6
    delta = 200
    for n in range(target, target + delta):
        if gcd(n, 10) == 1:
            if (av := a(n)):
                if av > target:
                    return n
    return None


if __name__ == "__main__":
    solution = euler_129()
    print("e129.py: " + str(solution))
    assert(solution == 1000023)