def p(leading_digits, count):
    n_cutoff = 18
    leading_digits_str = str(leading_digits)
    n_digits = len(leading_digits_str)
    j, current_leading, found = 1, 2, 0

    while found < count:
        current_leading *= 2
        j += 1
        current_leading_str = str(current_leading)
        if  current_leading_str[:n_digits] == leading_digits_str:
            found += 1
        current_leading = int(current_leading_str[:n_cutoff])
    return j


def euler_686():
    assert p(12, 1) == 7
    assert p(12, 2) == 80
    assert p(123, 45) == 12710
    assert p(123, 100) == 28343
    assert p(123, 1000) == 284168
    return p(123, 678910)


if __name__ == "__main__":
    solution = euler_686()
    print("e686.py: " + str(solution))
    assert solution == 193060223