Python/maths/sieve_of_eratosthenes.py

"""
Sieve of Eratosthones

The sieve of Eratosthenes is an algorithm used to find prime numbers, less than or equal to a given value.
Illustration: https://upload.wikimedia.org/wikipedia/commons/b/b9/Sieve_of_Eratosthenes_animation.gif
Reference: https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes

doctest provider: Bruno Simas Hadlich (https://github.com/brunohadlich)
Also thanks Dmitry (https://github.com/LizardWizzard) for finding the problem
"""


import math


def sieve(n):
    """
    Returns a list with all prime numbers up to n.

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

    l = [True] * (n + 1)  # noqa: E741
    prime = []
    start = 2
    end = int(math.sqrt(n))

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

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

        start += 1

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

    return prime


if __name__ == "__main__":
    print(sieve(int(input("Enter n: ").strip())))