Add optimized Sieve of Eratosthenes algorithm

maths/optimised_sieve_of_eratosthenes.py

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+# Optimized Sieve of Eratosthenes: An efficient algorithm to compute all prime numbers up to n.
+# 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).
+# Wikipedia URL - https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes
+
+from math import isqrt
+
+def optimized_sieve(n: int) -> list[int]:
+ """
+ Compute all prime numbers up to and including n using an optimized Sieve of Eratosthenes.
+
+ This implementation skips even numbers after 2 to reduce memory and runtime by about 50%.
+
+ Parameters
+ ----------
+ n : 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.
+
+ Returns
+ -------
+ list[int]
+ A list of primes in ascending order that are <= n.
+
+ Examples
+ --------
+ >>> optimized_sieve(10)
+ [2, 3, 5, 7]
+ >>> optimized_sieve(1)
+ []
+ >>> optimized_sieve(2)
+ [2]
+ >>> optimized_sieve(30)
+ [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
+ """
+ if n < 2:
+ return []
+
+ # Handle 2 separately, then consider only odd numbers
+ primes = [2] if n >= 2 else []
+
+ # Only odd numbers from 3 to n
+ size = (n - 1) // 2
+ is_prime = [True] * (size + 1)
+ limit = isqrt(n)
+
+ for i in range((limit - 1) // 2 + 1):
+ if is_prime[i]:
+ p = 2 * i + 3
+ # Start marking from p^2, converted to index
+ start = (p * p - 3) // 2
+ for j in range(start, size + 1, p):
+ is_prime[j] = False
+
+ primes.extend(2 * i + 3 for i in range(size + 1) if is_prime[i])
+ return primes
+
+
+if __name__ == "__main__":
+
+ print(optimized_sieve(50))