Sieve of Eratosthenes in Python

from bisect import bisect_left

# Sieve of Eratosthenes in Python
#
# PURPOSE
# The sieve identifies all primes up to a limit by marking multiples of smaller primes. This
# example returns their count.
#
# METHOD
# Keep a composite marker array. For each unmarked p up to the square root, mark multiples
# starting at p squared; smaller multiples were already handled. Count the remaining
# unmarked values from two onward.
#
# COMPLEXITY
# O(n log log n) time and O(n) auxiliary space.
#
# CONTRACT
# n >= 0. Returns the number of primes <= n. The marker array requires memory proportional
# to n.
# Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
# counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
# Sorted order, correct tree topology, and caller-owned data-structure invariants are
# preconditions; checking them on every query would change query complexity.
# Scalar and shape checks are included. Value-range checks are included where needed to
# prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
# workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
# uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
# partly updated after arithmetic failure. These are single-threaded, in-memory
# implementations.
#
# LANGUAGE DETAILS
# Python 3.9+: automatic lengths, native lists, arbitrary-precision arithmetic, and
# bisect_left for LIS. Invalid scalar/shape inputs raise ValueError. Integer element types
# and sorted/tree invariants remain caller preconditions.
#
# EXAMPLE
# Expected return: 4.
# Run: python example.py
# Tests for this exact source artifact run before publication. The accompanying source
# archive includes unit cases, runtime results, and source hashes.

def alg_require(condition):
    if not condition:
        raise ValueError("invalid algorithm input; see contract")
def alg_size(n):
    alg_require(0 <= n <= 4000000)
    return n

def sieve_count(n: int) -> int:
    alg_require(type(n) is int and -2147483648 <= n <= 2147483647)
    alg_require(n >= 0 and n <= 1000000)
    composite: list[int] = [0] * alg_size(n + 1)
    p: int = 2
    multiple: int = 0
    count: int = 0
    while p <= n // p:
        if composite[p] == 0:
            multiple = p * p
            while multiple <= n:
                composite[multiple] = 1
                multiple += p
        p += 1
    p = 2
    while p <= n:
        if composite[p] == 0:
            count += 1
        p += 1
    return count

n = 10
result = sieve_count(n)
print(result)