Sieve_of_sundaram

maths/sieve_of_sundaram.py

@@ -0,0 +1,89 @@
+"""
+Sieve of Sundaram Algorithm
+
+The Sieve of Sundaram is an algorithm for finding prime numbers up to a specified integer.
+It was discovered by Indian mathematician S. P. Sundaram in 1934.
+
+Wikipedia: https://en.wikipedia.org/wiki/Sieve_of_Sundaram
+"""
+
+from typing import List
+
+
+def sieve_of_sundaram(limit: int) -> List[int]:
+    """
+    Generate all prime numbers up to the given limit using Sieve of Sundaram.
+
+    The algorithm works by creating a list of integers and marking composite numbers,
+    then extracting the remaining unmarked numbers which represent primes.
+
+    Args:
+        limit: Upper bound (exclusive) for finding primes
+
+    Returns:
+        List of all prime numbers less than the limit
+
+    Raises:
+        ValueError: If limit is negative
+
+    Examples:
+        >>> sieve_of_sundaram(30)
+        [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
+        >>> sieve_of_sundaram(10)
+        [2, 3, 5, 7]
+        >>> sieve_of_sundaram(2)
+        []
+        >>> sieve_of_sundaram(4)
+        [2, 3]
+        >>> sieve_of_sundaram(1)
+        []
+        >>> sieve_of_sundaram(0)
+        []
+        >>> sieve_of_sundaram(-5)
+        Traceback (most recent call last):
+        ...
+        ValueError: limit must be non-negative
+    """
+    if limit < 0:
+        raise ValueError("limit must be non-negative")
+
+    if limit <= 2:
+        return []
+
+    # Calculate the range for Sundaram sieve
+    # We need to find primes up to 'limit', so we work with (limit-1)//2
+    n = (limit - 1) // 2
+
+    # Create a boolean array and initialize all entries as not marked (False)
+    # marked[i] represents whether (2*i + 1) is composite
+    marked = [False] * (n + 1)
+
+    # Mark numbers using Sundaram's formula: i + j + 2*i*j
+    # where i <= j and i + j + 2*i*j <= n
+    for i in range(1, n + 1):
+        j = i
+        while i + j + 2 * i * j <= n:
+            marked[i + j + 2 * i * j] = True
+            j += 1
+
+    # Collect unmarked numbers and transform them to get primes
+    primes = [2]  # 2 is the only even prime
+
+    for i in range(1, n + 1):
+        if not marked[i]:
+            primes.append(2 * i + 1)
+
+    return primes
+
+
+if __name__ == "__main__":
+    print("Sieve of Sundaram Demo")
+    print("-" * 20)
+
+    test_limits = [10, 30, 50, 100]
+
+    for limit in test_limits:
+        primes = sieve_of_sundaram(limit)
+        print(f"Primes up to {limit}: {primes}")
+        print(f"Count: {len(primes)}")
+        print()