import numpy as np

def sieve_np(limit):
    if limit < 2:
        return []

    # Create a boolean array "is_prime" and initialize all entries as True
    is_prime = np.ones(limit + 1, dtype=bool)
    is_prime[0:2] = False  # 0 and 1 are not prime numbers

    for p in range(2, int(limit**0.5) + 1):
        if is_prime[p]:
            # Use array slicing to mark all multiples of p as not prime
            is_prime[p**2 : limit + 1 : p] = False

    # Create a list of prime numbers using NumPy's where function
    primes = np.where(is_prime)[0]
    return primes.tolist()

def sieve(max_n):
    s = [True] * (max_n + 1)
    p = 2
    while p**2 <= max_n:
        if s[p]:
            for i in range(p**2, max_n + 1, p):
                s[i] = False
        p += 1
        
    primes = [p for p in range(2, max_n + 1) if s[p]]
    return primes


def sieve_of_eratosthenes(limit):
    if limit < 2:
        return []

    # Create a boolean array "prime[0..limit]" and initialize all entries as True
    prime = [True] * (limit + 1)
    prime[0] = prime[1] = False  # 0 and 1 are not prime numbers

    p = 2
    while p**2 <= limit:
        if prime[p]:
            # Update all multiples of p starting from p^2
            prime[p**2 : limit + 1 : p] = [False] * len(prime[p**2 : limit + 1 : p])
        p += 1

    # Create a list of prime numbers
    primes = [p for p in range(2, limit + 1) if prime[p]]
    return primes


class Primes:
    primes = sieve(25_000_000)
    
    @staticmethod
    def stream():
        yield from Primes.primes