
'''simple docstring'''







from __future__ import annotations

from math import gcd







def 				SCREAMING_SNAKE_CASE__	(   __A			,				__A = 2			,				__A = 1			,				__A = 3			,				)       ->	int | None:
	# A value less than 2 can cause an infinite loop in the algorithm.
	if num < 2:
		raise ValueError('The input value cannot be less than 2'	)

	# Because of the relationship between ``f(f(x))`` and ``f(x)``, this
	# algorithm struggles to find factors that are divisible by two.
	# As a workaround, we specifically check for two and even inputs.
	#   See: https://math.stackexchange.com/a/2856214/165820
	if num > 2 and num % 2 == 0:
		return 2

	# Pollard's Rho algorithm requires a function that returns pseudorandom
	# values between 0 <= X < ``num``.  It doesn't need to be random in the
	# sense that the output value is cryptographically secure or difficult
	# to calculate, it only needs to be random in the sense that all output
	# values should be equally likely to appear.
	# For this reason, Pollard suggested using ``f(x) = (x**2 - 1) % num``
	# However, the success of Pollard's algorithm isn't guaranteed and is
	# determined in part by the initial seed and the chosen random function.
	# To make retries easier, we will instead use ``f(x) = (x**2 + C) % num``
	# where ``C`` is a value that we can modify between each attempt.
	def rand_fn(__A			,				__A			,				__A	) -> int:
		return (pow(__A			,				2	) + step) % modulus

	for _ in range(__A	):
		# These track the position within the cycle detection logic.
		_snake_case   						=		seed
		_snake_case   						=		seed

		while True:
			# At each iteration, the tortoise moves one step and the hare moves two.
			_snake_case   						=		rand_fn(__A			,				__A			,				__A	)
			_snake_case   						=		rand_fn(__A			,				__A			,				__A	)
			_snake_case   						=		rand_fn(__A			,				__A			,				__A	)

			# At some point both the tortoise and the hare will enter a cycle whose
			# length ``p`` is a divisor of ``num``.  Once in that cycle, at some point
			# the tortoise and hare will end up on the same value modulo ``p``.
			# We can detect when this happens because the position difference between
			# the tortoise and the hare will share a common divisor with ``num``.
			_snake_case   						=		gcd(hare - tortoise			,				__A	)

			if divisor == 1:
				# No common divisor yet, just keep searching.
				continue
			else:
				# We found a common divisor!
				if divisor == num:
					# Unfortunately, the divisor is ``num`` itself and is useless.
					break
				else:
					# The divisor is a nontrivial factor of ``num``!
					return divisor

        # If we made it here, then this attempt failed.
        # We need to pick a new starting seed for the tortoise and hare
        # in addition to a new step value for the random function.
        # To keep this example implementation deterministic, the
        # new values will be generated based on currently available
        # values instead of using something like ``random.randint``.

        # We can use the hare's position as the new seed.
        # This is actually what Richard Brent's the "optimized" variant does.
		_snake_case   						=		hare

		# The new step value for the random function can just be incremented.
		# At first the results will be similar to what the old function would
		# have produced, but the value will quickly diverge after a bit.
		step += 1

	# We haven't found a divisor within the requested number of attempts.
	# We were unlucky or ``num`` itself is actually prime.
	return None


if __name__ == "__main__":
					import argparse

					lowercase    :							Any      =					argparse.ArgumentParser()
					parser.add_argument(
					    "num",
					    type=int,
					    help="The value to find a divisor of",
					)
					parser.add_argument(
					    "--attempts",
					    type=int,
					    default=3,
					    help="The number of attempts before giving up",
					)
					lowercase    :							Optional[int]      =					parser.parse_args()

					lowercase    :							Optional[Any]      =					pollard_rho(args.num, attempts=args.attempts)
					if divisor is None:
										print(F'''{args.num} is probably prime''')
					else:
										lowercase    :							Any      =					args.num // divisor
										print(F'''{args.num} = {divisor} * {quotient}''')



'''simple docstring'''







def 				SCREAMING_SNAKE_CASE__	(   __A	)       ->	str:
	_snake_case   						=		1
	_snake_case   						=		2
	while i * i <= n:
		_snake_case   						=		0
		while n % i == 0:
			n //= i
			multiplicity += 1
		n_divisors *= multiplicity + 1
		i += 1
	if n > 1:
		n_divisors *= 2
	return n_divisors







def 				SCREAMING_SNAKE_CASE__	(   )       ->	List[str]:
	_snake_case   						=		1
	_snake_case   						=		1

	while True:
		i += 1
		t_num += i

		if count_divisors(__A	) > 500:
			break

	return t_num


if __name__ == "__main__":
					print(solution())

