using filter and generator to generator endless prime number in python

Question

Below is a python program I found to find prime numbers using Sieve of Eratosthenes. It uses filter and generator. I\'m not able to understand it.

def _odd_i         


        

Answer 1

It's not just that single line of code, it's that line being run repeatedly, with different values of n.

Basically, it is an iterator that yields candidate prime numbers which have not yet been ruled out by the sieve. You start by making all odd numbers candidates.

it = _odd_iter()

Then you repeatedly take the first remaining candidate,

while True:
    n = next(it)

remove all numbers that are multiples of that candidate,

    filter(_not_divisible(n), it)

and replace your candidate primes with everything that is left after removing multiples.

    it = ...

If you pretend filter returns a list of numbers, rather than an iterable, and also pretend _odd_iter() returns a list of odd numbers instead of an iterable, you can trace through the loop and determine what's in the list at each point. For example, after running

it = _odd_iter()

you start with

it = 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, ...

Then run

    n = next(it) # 3

which pulls the first item off the front, leaving you with

it = 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, ...

and run

    it = filter(_not_divisible(3), it)

filter out all the multiples of 3,

it = 5, 7, 11, 13, 17, 19, 23, 25, ...

Then go back to the top of the loop and pull the new first number off the front

    n = next(it) # 5

leaving

it = 7, 11, 13, 17, 19, 23, 25, ...

and then filter out all the multiples of 5,

    it = filter(_not_divisible(5), it)

which gives

it = 7, 11, 13, 17, 19, 23, ...

and so on.

In practice, because filter() returns an iterator, not a list, you wind up getting a nested sequence of iterators. In particular, you start with

it = _odd_iter()

then after the first iteration of the loop, you have basically

it = filter(_non_divisible(3), _odd_iter())

except that 3 has been taken from the iterator, and then after the second iteration of the loop you have

it = filter(_non_divisible(5), filter(_non_divisible(3), _odd_iter()))

except that 5 has also been taken from the iterator, and then

it = filter(_non_divisible(7), filter(_non_divisible(5), filter(_non_divisible(3), _odd_iter())))

and so on.