Question

I’m trying to implement prime number generator using Sieve of Eratosthenes algorithm. I do it just to try using recursive iterator merging to implement sifter.

What I do is this:

from itertools import count,islice,groupby
from heapq import merge


def primes3():
    p = 2
    yield p
    sifter = (i*p for i in count(p))
    s = next(sifter)
    for p in count(p+1):
        if p==s: # this p is sieved out
            print('s: {}'.format(s))
            s = next(sifter)
        else:
            yield p # this is prime
            print('p: {}'.format(p))
            sifter = (k for k, g in groupby(merge(sifter,(i*p for i in count(p))))) #add this serie to the sifter: p*p, p*(p+1), p*(p+2), ...


print(list(islice(primes3(),10)))

The output is:

p: 3
s: 4
p: 5
p: 6
p: 7
p: 8
p: 9
p: 10
p: 11
s: 12
[2, 3, 5, 6, 7, 8, 9, 10, 11, 13]

The first number to be sieved out is 4. But the next is 12, not 6 as it should be. Why is that? I checked it with the following code:

>>> sifter = (i*2 for i in count(2))
>>> next(sifter)
4
>>> sifter = (k for k, g in groupby(merge(sifter,(i*3 for i in count(3)))))
>>> print(list(islice(sifter,20)))
[6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34]

So, as we may see, in test conditions sifter yields the correct results.

Where am I making a mistake? I think it must be something trivial that I just don’t see.

Answer 1

I have to agree, this stuff can sometimes be very confusing (but a nice puzzle!).

Turns out that your sifter iterator changes when the value of p changes (by the way, I’m using python 2.6.5 to test this, not python 3, so print syntax is a bit different):

>> p = 2
>> sifter = (i*p for i in count(p))
>> for x in range(3):
>>    print next(sifter)
4
6
8
>>> # now lets see what happens when we change p
>>> p = 3
>>> for x in range(3):
>>>     print next(sifter)
15
18
21

So, the i*p part of the iterator is evaluated with the new of p as soon as p has been updated. An p is indeed updated in your mainloop, which is why you don’t get the expected result.

There is an easy way to solve this: create a function to generate the sifter iterator such that the iterator isn’t bounded to p:

def gensifter(x):
    return (i*x for i in count(x))

and put this in your code (again, I converted it to python 2.6.5):

def primes3():
    p = 2
    yield p
    sifter = gensifter(p) # <-- changed!
    s = next(sifter)
    for p in count(p+1):
        #print '(testing p = %d\ts = %d)' % (p, s)
        if p==s: # this p is sieved out
            print 's:', s
            s = next(sifter)
        else:
            yield p # this is prime
            print 'p:', p
            sifter = (k for k, g in groupby(merge(sifter,gensifter(p)))) # <-- changed!

Let’s see the result now:

>>> print list(islice(primes3(), 10))
p: 3
s: 4
p: 5
s: 6
p: 7
s: 8
s: 9
s: 10
p: 11
s: 12
p: 13
s: 14
s: 15
s: 16
p: 17
s: 18
p: 19
s: 20
s: 21
s: 22
p: 23
s: 24
s: 25
s: 26
s: 27
s: 28
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]

Primenumbers galore!