[mypy] math/sieve_of_eratosthenes: Add type hints (#2627)

* add type hints to math/sieve
* add doctest
* math/sieve: remove manual doctest
* add check for negative
* Update maths/sieve_of_eratosthenes.py
* Update sieve_of_eratosthenes.py

Co-authored-by: Dhruv Manilawala <dhruvmanila@gmail.com>

maths/sieve_of_eratosthenes.py

@@ -8,54 +8,58 @@ https://upload.wikimedia.org/wikipedia/commons/b/b9/Sieve_of_Eratosthenes_animat
 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
+Also thanks to Dmitry (https://github.com/LizardWizzard) for finding the problem
 """
 
 
 import math
+from typing import List
 
 
-def sieve(n):
+def prime_sieve(num: int) -> List[int]:
     """
     Returns a list with all prime numbers up to n.
 
-    >>> sieve(50)
+    >>> prime_sieve(50)
     [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
-    >>> sieve(25)
+    >>> prime_sieve(25)
     [2, 3, 5, 7, 11, 13, 17, 19, 23]
-    >>> sieve(10)
+    >>> prime_sieve(10)
     [2, 3, 5, 7]
-    >>> sieve(9)
+    >>> prime_sieve(9)
     [2, 3, 5, 7]
-    >>> sieve(2)
+    >>> prime_sieve(2)
     [2]
-    >>> sieve(1)
+    >>> prime_sieve(1)
     []
     """
 
-    l = [True] * (n + 1)  # noqa: E741
+    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(n))
+    end = int(math.sqrt(num))
 
     while start <= end:
         # If start is a prime
-        if l[start] is True:
+        if sieve[start] is True:
             prime.append(start)
 
             # Set multiples of start be False
-            for i in range(start * start, n + 1, start):
+            for i in range(start * start, num + 1, start):
-                if l[i] is True:
+                if sieve[i] is True:
-                    l[i] = False
+                    sieve[i] = False
 
         start += 1
 
-    for j in range(end + 1, n + 1):
+    for j in range(end + 1, num + 1):
-        if l[j] is True:
+        if sieve[j] is True:
             prime.append(j)
 
     return prime
 
 
 if __name__ == "__main__":
-    print(sieve(int(input("Enter n: ").strip())))
+    print(prime_sieve(int(input("Enter a positive integer: ").strip())))