Added odd_sieve.py (#8740)

maths/odd_sieve.py

@@ -0,0 +1,42 @@
+from itertools import compress, repeat
+from math import ceil, sqrt
+
+
+def odd_sieve(num: int) -> list[int]:
+    """
+    Returns the prime numbers < `num`. The prime numbers are calculated using an
+    odd sieve implementation of the Sieve of Eratosthenes algorithm
+    (see for reference https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes).
+
+    >>> odd_sieve(2)
+    []
+    >>> odd_sieve(3)
+    [2]
+    >>> odd_sieve(10)
+    [2, 3, 5, 7]
+    >>> odd_sieve(20)
+    [2, 3, 5, 7, 11, 13, 17, 19]
+    """
+
+    if num <= 2:
+        return []
+    if num == 3:
+        return [2]
+
+    # Odd sieve for numbers in range [3, num - 1]
+    sieve = bytearray(b"\x01") * ((num >> 1) - 1)
+
+    for i in range(3, int(sqrt(num)) + 1, 2):
+        if sieve[(i >> 1) - 1]:
+            i_squared = i**2
+            sieve[(i_squared >> 1) - 1 :: i] = repeat(
+                0, ceil((num - i_squared) / (i << 1))
+            )
+
+    return [2] + list(compress(range(3, num, 2), sieve))
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()