



import math
from collections.abc import Generator


def 						UpperCAmelCase__   (   UpperCAmelCase__			:int       ):






 '''simple docstring'''

 a							=		(i for i in range(1			,       (max_n + 1)       ))
 for i in (n for n in numbers if n > 1):
  for j in range(2			,       UpperCAmelCase__       ):
   if (i % j) == 0:
    break
  else:
   yield i


def 						UpperCAmelCase__   (   UpperCAmelCase__			:int       ):






 '''simple docstring'''

 a							=		(i for i in range(1			,       (max_n + 1)       ))
 for i in (n for n in numbers if n > 1):
  # only need to check for factors up to sqrt(i)
  a							=		int(math.sqrt(UpperCAmelCase__       )       ) + 1
  for j in range(2			,       UpperCAmelCase__       ):
   if (i % j) == 0:
    break
  else:
   yield i


def 						UpperCAmelCase__   (   UpperCAmelCase__			:int       ):






 '''simple docstring'''

 a							=		(i for i in range(1			,       (max_n + 1)			,       2       ))
 # It's useless to test even numbers as they will not be prime
 if max_n > 2:
  yield 2  # Because 2 will not be tested, it's necessary to yield it now
 for i in (n for n in numbers if n > 1):
  a							=		int(math.sqrt(UpperCAmelCase__       )       ) + 1
  for j in range(3			,       UpperCAmelCase__			,       2       ):
   # As we removed the even numbers, we don't need them now
   if (i % j) == 0:
    break
  else:
   yield i


def 						UpperCAmelCase__   (   ):






 '''simple docstring'''

 from timeit import timeit

 a							=		"from __main__ import slow_primes, primes, fast_primes"
 print(timeit("slow_primes(1_000_000_000_000)"			,       setup=UpperCAmelCase__			,       number=1_00_00_00       )       )
 print(timeit("primes(1_000_000_000_000)"			,       setup=UpperCAmelCase__			,       number=1_00_00_00       )       )
 print(timeit("fast_primes(1_000_000_000_000)"			,       setup=UpperCAmelCase__			,       number=1_00_00_00       )       )


if __name__ == "__main__":
     A_		:  Tuple										=  int(input('''Calculate primes up to:\n>> ''').strip())
     for ret in primes(number):
          print(ret)
     benchmark()






from __future__ import annotations


def 						UpperCAmelCase__   (   UpperCAmelCase__			:float | int			,       UpperCAmelCase__			:float | int			,       UpperCAmelCase__			:float | int			,       ):






 '''simple docstring'''

 if not all((nth_term, start_term_a, common_ratio_r)       ):
  return []
 a							=		[]
 a							=		1
 a							=		common_ratio_r
 for _ in range(int(UpperCAmelCase__       )       ):
  if not series:
   series.append(UpperCAmelCase__       )
  else:
   power += 1
   series.append(float(start_term_a * multiple       )       )
   a							=		pow(float(UpperCAmelCase__       )			,       UpperCAmelCase__       )
 return series


if __name__ == "__main__":
     import doctest

     doctest.testmod()

     A_		:  Optional[Any]										=  float(input('''Enter the last number (n term) of the Geometric Series'''))
     A_		:  Optional[Any]										=  float(input('''Enter the starting term (a) of the Geometric Series'''))
     A_		:  Optional[int]										=  float(
         input('''Enter the common ratio between two terms (r) of the Geometric Series''')
     )
     print('''Formula of Geometric Series => a + ar + ar^2 ... +ar^n''')
     print(geometric_series(nth_term, start_term_a, common_ratio_r))

