ROMAN = [
(1000000, "M_"), (900000, "C_M_"), (500000, "D_"), (400000, "C_D_"),
    (100000, "C_"), (90000, "X_C_"), (50000, "L_"), (40000, "X_L_"),
    (10000, "X_"), (9000, "I_X_"), (5000, "V_"), (4000, "I_V_"),
    (1000, "M"), (900, "CM"), (500, "D"), (400, "CD"),
    (100, "C"), (90, "XC"), (50, "L"), (40, "XL"),
    (10, "X"), (9, "IX"), (5, "V"), (4, "IV"), (1, "I")
]
def roman_to_int(roman):
    """
    Convert a Roman numeral to an integer, supporting Vinculum notation
    (underscore _ represents 1000 times).

    LeetCode No. 13 Roman to Integer:
    Given a Roman numeral, convert it to an integer.
    Input is guaranteed to be within the range from 1 to 3999.

    Reference: https://en.wikipedia.org/wiki/Roman_numerals
    >>> tests = {"III": 3, "CLIV": 154, "MIX": 1009, "MMD": 2500,
    ...          "MMMCMXCIX": 3999, "I_V_": 4000, "X_": 10000, "M_": 1000000}
    >>> all(roman_to_int(key) == value for key, value in tests.items())
    True
    """
    vals = {roman: arabic for arabic, roman in ROMAN}
    i, total = 0, 0
    while i < len(roman):
    if i + 1 < len(roman) and roman[i + 1] == "_":
        total += vals[roman[i] + "_"]
        i += 2
    else:
        total += vals[roman[i]]
        i += 1
def int_to_roman(number):
    """
    Convert an integer to a Roman numeral, supporting Vinculum notation
    (underscore _ represents 1000 times).

    Given an integer, convert it to a Roman numeral.

    Reference: https://en.wikipedia.org/wiki/Roman_numerals
    >>> tests = {3: "III", 154: "CLIV", 1009: "MIX", 2500: "MMD", 3999: "MMMCMXCIX"}
    >>> all(int_to_roman(value) == key for key, value in tests.items())
    True
    """
    if not isinstance(number, int) or number <= 0:
        raise ValueError("Input must be a positive integer greater than 0")

    result = []
    for arabic, roman in ROMAN:
        factor, number = divmod(number, arabic)
        result.append(roman * factor)
        if number == 0:
            break
    return "".join(result)

if __name__ == "__main__":
    import doctest
    doctest.testmod()