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_iter():
    n = 1
    while True:
        n = n + 2
        yield n

def _not_divisible(n):
    return lambda x: x % n > 0

def primes():
    yield 2
    it = _odd_iter()
    while True:
        n = next(it)
        yield n
        it = filter(_not_divisible(n), it)

for n in primes():
    if n < 1000:
        print(n)
    else:
        break

What I don't understand is it = filter(_not_divisible(n), it). For example for number 105, how is it excluded by this single line of code?

Answer 1

For each prime number found a filter is applied to the iterable, the filter used is a function that excludes all multiples of the prime number.

So your iterable is wrapped in as many filters as you found prime numbers, for example the number 105 is excluded because it's divisible by 3 and the filter for all multiples of 3 was added when you found the prime number 3.

If you add some print statements it will be a bit clearer (I hope):

def _odd_iter():
    n = 1
    while True:
        n = n + 2
        yield n

def _not_divisible(n):
    print('add filter for all multiples of', n)
    return lambda x: print('check if', x, 'is divisible by', n, 'result: ', not (x % n > 0)) or x % n > 0

def primes():
    yield 2
    it = _odd_iter()
    while True:
        n = next(it)
        yield n
        it = filter(_not_divisible(n), it)

for n in primes():
    if n < 20:
        print(n)
    else:
        break

which prints:

2
3
add filter for all multiples of 3
check if 5 is divisible by 3 result:  False
5
add filter for all multiples of 5
check if 7 is divisible by 3 result:  False
check if 7 is divisible by 5 result:  False
7
add filter for all multiples of 7
check if 9 is divisible by 3 result:  True
check if 11 is divisible by 3 result:  False
check if 11 is divisible by 5 result:  False
check if 11 is divisible by 7 result:  False
11
add filter for all multiples of 11
check if 13 is divisible by 3 result:  False
check if 13 is divisible by 5 result:  False
check if 13 is divisible by 7 result:  False
check if 13 is divisible by 11 result:  False
13
add filter for all multiples of 13
check if 15 is divisible by 3 result:  True
check if 17 is divisible by 3 result:  False
check if 17 is divisible by 5 result:  False
check if 17 is divisible by 7 result:  False
check if 17 is divisible by 11 result:  False
check if 17 is divisible by 13 result:  False
17
add filter for all multiples of 17
check if 19 is divisible by 3 result:  False
check if 19 is divisible by 5 result:  False
check if 19 is divisible by 7 result:  False
check if 19 is divisible by 11 result:  False
check if 19 is divisible by 13 result:  False
check if 19 is divisible by 17 result:  False
19
add filter for all multiples of 19
check if 21 is divisible by 3 result:  True
check if 23 is divisible by 3 result:  False
check if 23 is divisible by 5 result:  False
check if 23 is divisible by 7 result:  False
check if 23 is divisible by 11 result:  False
check if 23 is divisible by 13 result:  False
check if 23 is divisible by 17 result:  False
check if 23 is divisible by 19 result:  False