
"""simple docstring"""




def 							a__ (				lowerCAmelCase	:       int = 100_0000		):






     '''simple docstring'''
     UpperCAmelCase__							: List[str] 							=						set(range(3       ,    lowerCAmelCase       ,    2		)		)
     primes.add(2		)
     for p in range(3       ,    lowerCAmelCase       ,    2		):
          if p not in primes:
               continue
          primes.difference_update(set(range(p * p       ,    lowerCAmelCase       ,    lowerCAmelCase		)		)		)

     UpperCAmelCase__							: int 							=						[float(lowerCAmelCase		) for n in range(limit + 1		)]

     for p in primes:
          for n in range(lowerCAmelCase       ,    limit + 1       ,    lowerCAmelCase		):
               phi[n] *= 1 - 1 / p

     return int(sum(phi[2:]		)		)


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





"""simple docstring"""




import math



def 							a__ (				lowerCAmelCase	:       int		):






     '''simple docstring'''

     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(lowerCAmelCase		) + 1		)       ,    6		):
          if number % i == 0 or number % (i + 2) == 0:
               return False
     return True



def 							a__ (				lowerCAmelCase	:       int = 1_0001		):






     '''simple docstring'''

     try:
          UpperCAmelCase__							: List[str] 							=						int(lowerCAmelCase		)
     except (TypeError, ValueError):
          raise TypeError("Parameter nth must be int or castable to int."		) from None
     if nth <= 0:
          raise ValueError("Parameter nth must be greater than or equal to one."		)
     UpperCAmelCase__							: list[int] 							=						[]
     UpperCAmelCase__							: str 							=						2
     while len(lowerCAmelCase		) < nth:
          if is_prime(lowerCAmelCase		):
               primes.append(lowerCAmelCase		)
               num += 1
          else:
               num += 1
     return primes[len(lowerCAmelCase		) - 1]


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



