





def   a_    ( lowerCAmelCase_      :			int	):

   if num <= 0:
      raise ValueError('Input must be a positive integer'	)

   __lowerCAmelCase								=	[True] * (num + 1)

   __lowerCAmelCase								=	2
   while p * p <= num:
      if primes[p]:
         for i in range(p * p,	num + 1,	lowerCAmelCase_	):
            __lowerCAmelCase								=	False
      p += 1

   return [prime for prime in range(2,	num + 1	) if primes[prime]]


if __name__ == "__main__":
     import doctest

     doctest.testmod()

     _snake_case  :    Optional[Any]						    =  int(input('Enter a positive integer: ').strip())
     print(prime_sieve_eratosthenes(user_num))








def   a_    ( lowerCAmelCase_      :			int	):

   if p < 2:
      raise ValueError('p should not be less than 2!'	)
   elif p == 2:
      return True

   __lowerCAmelCase								=	4
   __lowerCAmelCase								=	(1 << p) - 1
   for _ in range(p - 2	):
      __lowerCAmelCase								=	((s * s) - 2) % m
   return s == 0


if __name__ == "__main__":
     print(lucas_lehmer_test(7))
     print(lucas_lehmer_test(11))

