import math


def prime_sieve(num: int) -> list[int]:
    """
    Returns a list with all prime numbers up to n.

    >>> prime_sieve(50)
    [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
    >>> prime_sieve(25)
    [2, 3, 5, 7, 11, 13, 17, 19, 23]
    >>> prime_sieve(10)
    [2, 3, 5, 7]
    >>> prime_sieve(9)
    [2, 3, 5, 7]
    >>> prime_sieve(2)
    [2]
    >>> prime_sieve(1)
    []
    """

    if num <= 0:
        raise ValueError(f"{num}: Invalid input, please enter a positive integer.")

    sieve = [True] * (num + 1)
    prime = []
    start = 2
    end = int(math.sqrt(num))

    while start <= end:
        # If start is a prime
        if sieve[start] is True:
            prime.append(start)

            # Set multiples of start be False
            for i in range(start * start, num + 1, start):
                if sieve[i] is True:
                    sieve[i] = False

        start += 1

    for j in range(end + 1, num + 1):
        if sieve[j] is True:
            prime.append(j)

    return prime