#7. Algorithm for determining all Carmichael numbers less than a given bound.

import math

# Helper function to find all prime factors of a number
def prime_factors(n):
    factors = set()
    # Check for divisibility by 2
    while n % 2 == 0:
        factors.add(2)
        n //= 2
    # Check for odd factors from 3 upwards
    for i in range(3, int(math.sqrt(n)) + 1, 2):
        while n % i == 0:
            factors.add(i)
            n //= i
    # If n is prime and greater than 2 then add it to the set
    if n > 2:
        factors.add(n)
    return factors

# Function to check if a number is square-free (no repeated prime factors)
def is_square_free(n, prime_factors):
    for p in prime_factors:
        if n % (p * p) == 0:
            return False
    return True

# Function to check for Carmichael numbers
def is_carmichael(n):
    prime_factors_of_n = prime_factors(n)
    if(len(prime_factors_of_n)<2):
        return False
    if n < 3 or not is_square_free(n, prime_factors_of_n):
        return False
    for p in prime_factors_of_n:
        if (n - 1) % (p - 1) != 0:
            return False
    return True

# Function to find all Carmichael numbers below a given bound
def carmichael_numbers_below(bound):
    carmichaels = []
    for n in range(3, bound):
        if is_carmichael(n):
            carmichaels.append(n)
    return carmichaels

bound = 552722
carmichaels = carmichael_numbers_below(bound)
print(f"Carmichael numbers below {bound}:")
print(carmichaels)

#https://oeis.org/A002997