feat: add sieve of Sundaram

Author: peeyushtiwary27

maths/sieve_of_sundaram.py

@@ -0,0 +1,67 @@
+"""
+Sieve of Sundaram algorithm for finding prime numbers up to a given limit.
+
+The Sieve of Sundaram generates prime numbers by eliminating integers
+of the form i + j + 2ij.
+
+Time Complexity: O(n log n)
+Space Complexity: O(n)
+
+Reference: https://en.wikipedia.org/wiki/Sieve_of_Sundaram
+"""
+
+
+def sieve_of_sundaram(limit: int) -> list[int]:
+    """
+    Generate all prime numbers up to a given limit.
+
+    Args:
+        limit: Upper bound for finding primes (inclusive).
+
+    Returns:
+        List of prime numbers up to the given limit.
+
+    Raises:
+        ValueError: If limit is negative.
+
+    Examples:
+        >>> sieve_of_sundaram(20)
+        [2, 3, 5, 7, 11, 13, 17, 19]
+        >>> sieve_of_sundaram(10)
+        [2, 3, 5, 7]
+        >>> sieve_of_sundaram(2)
+        [2]
+        >>> sieve_of_sundaram(1)
+        []
+        >>> sieve_of_sundaram(0)
+        []
+        >>> sieve_of_sundaram(-1)
+        Traceback (most recent call last):
+        ...
+        ValueError: -1: Invalid input, please enter a non-negative integer.
+    """
+    if limit < 0:
+        msg = f"{limit}: Invalid input, please enter a non-negative integer."
+        raise ValueError(msg)
+
+    if limit < 2:
+        return []
+
+    upper_bound = (limit - 1) // 2
+    marked = [False] * (upper_bound + 1)
+
+    for i in range(1, upper_bound + 1):
+        j = i
+        while i + j + 2 * i * j <= upper_bound:
+            marked[i + j + 2 * i * j] = True
+            j += 1
+
+    primes = [2]
+
+    for value in range(1, upper_bound + 1):
+        if not marked[value]:
+            prime = 2 * value + 1
+            if prime <= limit:
+                primes.append(prime)
+
+    return primes