





"""simple docstring"""






from typing import List

import numpy as np

def       __A (     a_  :dict)						->						int:
				__a	:  Any         =					{key: len(a_) for key, value in gen_kwargs.items() if isinstance(a_				,    a_)}
				if len(set(lists_lengths.values())) > 1:
								raise RuntimeError(
								    (
								        '''Sharding is ambiguous for this dataset: '''
								        + '''we found several data sources lists of different lengths, and we don\'t know over which list we should parallelize:\n'''
								        + '''\n'''.join(F"""\t- key {key} has length {length}""" for key, length in lists_lengths.items())
								        + '''\nTo fix this, check the \'gen_kwargs\' and make sure to use lists only for data sources, '''
								        + '''and use tuples otherwise. In the end there should only be one single list, or several lists with the same length.'''
								    ))
				__a	:  List[str]         =					max(lists_lengths.values()				,    default=0)
				return max(1				,    a_)

def       __A (     a_  :int				,    a_  :int)						->						List[range]:
				__a	:  int         =					[]
				for group_idx in range(a_):
								__a	:  Optional[Any]         =					num_shards // max_num_jobs + (group_idx < (num_shards % max_num_jobs))
								if num_shards_to_add == 0:
												break
								__a	:  Dict         =					shards_indices_per_group[-1].stop if shards_indices_per_group else 0
								__a	:  List[Any]         =					range(a_				,    start + num_shards_to_add)
								shards_indices_per_group.append(a_)
				return shards_indices_per_group

def       __A (     a_  :dict				,    a_  :int)						->						List[dict]:
				__a	:  Any         =					_number_of_shards_in_gen_kwargs(a_)
				if num_shards == 1:
								return [dict(a_)]
				else:
								__a	:  int         =					_distribute_shards(num_shards=a_				,    max_num_jobs=a_)
								return [
								    {
								        key: [value[shard_idx] for shard_idx in shard_indices_per_group[group_idx]]
								        if isinstance(a_				,    a_)
								        else value
								        for key, value in gen_kwargs.items()
								    }
								    for group_idx in range(len(a_))
								]

def       __A (     a_  :List[dict])						->						dict:
				return {
				    key: [value for gen_kwargs in gen_kwargs_list for value in gen_kwargs[key]]
				    if isinstance(gen_kwargs_list[0][key]				,    a_)
				    else gen_kwargs_list[0][key]
				    for key in gen_kwargs_list[0]
				}






def       __A (     a_  :np.random.Generator				,    a_  :dict)						->						dict:
				__a	:  List[str]         =					{len(a_) for value in gen_kwargs.values() if isinstance(a_				,    a_)}
				__a	:  Optional[int]         =					{}
				for size in list_sizes:
								__a	:  str         =					list(range(a_))
								rng.shuffle(indices_per_size[size])
				# Now let's copy the gen_kwargs and shuffle the lists based on their sizes
				__a	:  Any         =					dict(a_)
				for key, value in shuffled_kwargs.items():
								if isinstance(a_				,    a_):
												__a	:  Any         =					[value[i] for i in indices_per_size[len(a_)]]
				return shuffled_kwargs






"""simple docstring"""






def       __A (     a_  :int = 1_00_00_00)						->						int:
				__a	:  Tuple         =					[i - 1 for i in range(limit + 1)]

				for i in range(2				,    limit + 1):
								if phi[i] == i - 1:
												for j in range(2 * i				,    limit + 1				,    a_):
																phi[j] -= phi[j] // i

				return sum(phi[2 : limit + 1])


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