Several example codes

Algorithm Overview

The Sieve of Eratosthenes is a classical and efficient method for finding all prime numbers up to a given number n.
It works by iteratively marking the multiples of each prime number starting from 2.

Steps:

1. Start from 2 (the first prime)
2. Remove all multiples of the current prime
3. Move to the next unmarked number and repeat
4. Stop when the current number exceeds √n

Python Code:

def sieve_of_eratosthenes(n: int) -> list[int]:
    is_prime = [True] * (n + 1)
    is_prime[0:2] = [False, False]  # 0 and 1 are not prime

    for i in range(2, int(n**0.5) + 1):
        if is_prime[i]:
            for multiple in range(i * i, n + 1, i):
                is_prime[multiple] = False

    return [i for i, prime in enumerate(is_prime) if prime]

# Run sieve up to 199
primes_up_to_199 = sieve_of_eratosthenes(199)
print("Prime numbers up to 199:")
print(primes_up_to_199)

Sample Output:

Prime numbers up to 199:  
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199]