from array import array
from collections.abc import Generator, Iterator
from typing import Any, Literal

from sympy import Function
from sympy.core.function import UndefinedFunction
from sympy.utilities.decorator import deprecated

class Sieve:
    def __init__(self) -> None: ...
    def extend(self, n) -> None: ...
    def extend_to_no(self, i) -> None: ...
    def primerange(self, a, b=...) -> Generator[Any, Any, None]: ...
    def totientrange(self, a, b) -> Generator[int, Any, None]: ...
    def mobiusrange(self, a, b) -> Generator[int, Any, None]: ...
    def search(self, n) -> tuple[int, int]: ...
    def __contains__(self, n) -> bool: ...
    def __iter__(self) -> Iterator[array[int] | int]: ...
    def __getitem__(self, n) -> array[int] | int: ...

sieve = ...

def prime(nth) -> array[int] | int: ...

class primepi(Function):
    @classmethod
    def eval(cls, n) -> None: ...

def nextprime(n, ith=...) -> int | array[int] | None: ...
def prevprime(n) -> int | array[int] | None: ...
def primerange(a, b=...) -> Generator[Any | int | array[int] | None, Any, None]: ...
def randprime(a, b) -> int | array[int] | None: ...
def primorial(n, nth=...) -> array[int] | int: ...
def cycle_length(f, x0, nmax=..., values=...) -> Generator[Any | tuple[int, None] | tuple[int, int], Any, None]: ...
def composite(nth) -> int: ...
def compositepi(n) -> Literal[0]: ...