Update optimised_sieve_of_eratosthenes.py

Author: Naman-Vasudev

maths/optimised_sieve_of_eratosthenes.py

@@ -1,26 +1,26 @@
-# Optimized Sieve of Eratosthenes: An efficient algorithm to compute all prime numbers up to n.
+# Optimized Sieve of Eratosthenes: An efficient algorithm to compute all prime numbers up to limit.
 # This version skips even numbers after 2, improving both memory and time usage.
-# It is particularly efficient for larger n (e.g., up to 10**8 on typical hardware).
+# It is particularly efficient for larger limits (e.g., up to 10**8 on typical hardware).
 # Wikipedia URL - https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes
 
 from math import isqrt
 
-def optimized_sieve(n: int) -> list[int]:
+def optimized_sieve(limit: int) -> list[int]:
     """
-    Compute all prime numbers up to and including n using an optimized Sieve of Eratosthenes.
+    Compute all prime numbers up to and including `limit` using an optimized Sieve of Eratosthenes.
 
     This implementation skips even numbers after 2 to reduce memory and runtime by about 50%.
 
     Parameters
     ----------
-    n : int
+    limit : int
         Upper bound (inclusive) of the range in which to find prime numbers.
-        Expected to be a non-negative integer. If n < 2 the function returns an empty list.
+        Expected to be a non-negative integer. If limit < 2 the function returns an empty list.
 
     Returns
     -------
     list[int]
-        A list of primes in ascending order that are <= n.
+        A list of primes in ascending order that are <= limit.
 
     Examples
     --------
@@ -33,18 +33,18 @@ def optimized_sieve(n: int) -> list[int]:
     >>> optimized_sieve(30)
     [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
     """
-    if n < 2:
+    if limit < 2:
         return []
 
     # Handle 2 separately, then consider only odd numbers
-    primes = [2] if n >= 2 else []
+    primes = [2] if limit >= 2 else []
 
-    # Only odd numbers from 3 to n
-    size = (n - 1) // 2
+    # Only odd numbers from 3 to limit
+    size = (limit - 1) // 2
     is_prime = [True] * (size + 1)
-    limit = isqrt(n)
+    bound = isqrt(limit)
 
-    for i in range((limit - 1) // 2 + 1):
+    for i in range((bound - 1) // 2 + 1):
         if is_prime[i]:
             p = 2 * i + 3
             # Start marking from p^2, converted to index
@@ -57,5 +57,4 @@ def optimized_sieve(n: int) -> list[int]:
 
 
 if __name__ == "__main__":
-
     print(optimized_sieve(50))