



a_       :Any				  =				{
    "0": 1,
    "1": 1,
    "2": 2,
    "3": 6,
    "4": 24,
    "5": 120,
    "6": 720,
    "7": 5_040,
    "8": 40_320,
    "9": 362_880,
}

a_       :Optional[Any]				  =				{145: 145}

a_       :Dict				  =				{
    145: 0,
    169: 3,
    36_301: 3,
    1_454: 3,
    871: 2,
    45_361: 2,
    872: 2,
}
def 	lowercase_							(A			:			int				):
       if n in CACHE_SUM_DIGIT_FACTORIALS:
              return CACHE_SUM_DIGIT_FACTORIALS[n]
       snake_case__							:     List[Any]            =     sum(DIGIT_FACTORIALS[let] for let in str(A				)				)
       snake_case__							:     Any            =     ret
       return ret
def 	lowercase_							(A			:			int		, A			:			set | None = None				):
       snake_case__							:     List[Any]            =     previous or set()
       if n in CHAIN_LENGTH_CACHE:
              return CHAIN_LENGTH_CACHE[n]
       snake_case__							:     Union[str, Any]            =     sum_digit_factorials(A				)
       if next_number in previous:
              snake_case__							:     str            =     0
              return 0
       else:
              previous.add(A				)
              snake_case__							:     Dict            =     1 + chain_length(A		, A				)
              snake_case__							:     Tuple            =     ret
              return ret
def 	lowercase_							(A			:			int = 6_0		, A			:			int = 1_0_0_0_0_0_0				):
       return sum(1 for i in range(1		, A				) if chain_length(A				) == num_terms				)


if __name__ == "__main__":
 print(F"""{solution() = }""")











import math
def 	lowercase_							(A			:			int				):

       if 1 < number < 4:
              # 2 and 3 are primes
              return True
       elif number < 2 or number % 2 == 0 or number % 3 == 0:
              # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
              return False

       # All primes number are in format of 6k +/- 1
       for i in range(5		, int(math.sqrt(A				) + 1				)		, 6				):
              if number % i == 0 or number % (i + 2) == 0:
                     return False
       return True
def 	lowercase_							(A			:			int = 1_0_0_0		, A			:			int = 1_0_0_0				):
       snake_case__							:     str            =     [0, 0, 0]  # length, a, b
       for a in range((a_limit * -1) + 1		, A				):
              for b in range(2		, A				):
                     if is_prime(A				):
                            snake_case__							:     Optional[Any]            =     0
                            snake_case__							:     Optional[int]            =     0
                            while is_prime((n**2) + (a * n) + b				):
                                   count += 1
                                   n += 1
                            if count > longest[0]:
                                   snake_case__							:     Union[str, Any]            =     [count, a, b]
       snake_case__							:     List[str]            =     longest[1] * longest[2]
       return ans


if __name__ == "__main__":
 print(solution(1_000, 1_000))






