How to implement an efficient infinite generator of prime numbers in Python?

Question

This is not a homework, I am just curious.

INFINITE is the key word here.

I wish to use it as for p in primes(). I believe that this is a built-in function in Haskell.

So, the answer cannot be as naive as "Just do a Sieve".

First of all, you do not know how many consecutive primes will be consumed. Well, suppose you could concoct 100 of them at a time. Would you use the same Sieve approach as well as the frequency of prime numbers formula?

I prefer non-concurrent approach.

Thank you for reading (and writing ;) )!

Answer 1

Since the OP asks for an efficient implementation, here's a significant improvement to the active state 2002 code by David Eppstein/Alex Martelli (seen here in his answer): don't record a prime's info in the dictionary until its square is seen among the candidates. Brings space complexity down to below O(sqrt(n)) instead of O(n), for n primes produced ( π(sqrt(n log n)) ~ 2 sqrt(n log n) / log(n log n) ~ 2 sqrt(n / log n) ). Consequently, time complexity is also improved, i.e. it runs faster.

Creates a "sliding sieve" as a dictionary of current multiples of each base prime (i.e. below the sqrt of the current production point), together with their step values:

from itertools import count
                                         # ideone.com/aVndFM
def postponed_sieve():                   # postponed sieve, by Will Ness      
    yield 2; yield 3; yield 5; yield 7;  # original code David Eppstein, 
    sieve = {}                           #   Alex Martelli, ActiveState Recipe 2002
    ps = postponed_sieve()               # a separate base Primes Supply:
    p = next(ps) and next(ps)            # (3) a Prime to add to dict
    q = p*p                              # (9) its sQuare 
    for c in count(9,2):                 # the Candidate
        if c in sieve:               # c's a multiple of some base prime
            s = sieve.pop(c)         #     i.e. a composite ; or
        elif c < q:  
             yield c                 # a prime
             continue              
        else:   # (c==q):            # or the next base prime's square:
            s=count(q+2*p,2*p)       #    (9+6, by 6 : 15,21,27,33,...)
            p=next(ps)               #    (5)
            q=p*p                    #    (25)
        for m in s:                  # the next multiple 
            if m not in sieve:       # no duplicates
                break
        sieve[m] = s                 # original test entry: ideone.com/WFv4f

(the older, original code here was edited to incorporate changes as seen in the answer by Tim Peters, below). see also this for a related discussion.

Similar 2-3-5-7 wheel-based code runs ~ 2.15x faster (which is very close to the theoretical improvement of 3/2 * 5/4 * 7/6 = 2.1875).

2022 update: I've recently chanced upon this old "NESL" thing from the 1990s which actually uses the same sqrt-recursion trick. So nothing is new under the sun. :)

The code can be straightforwardly augmented to start the primes enumeration directly from a given value. This can be seen in this JS-based entry.

Answer 2

I know the post is old, but I came across this question... The following code is based on a very simple idea: a growing sieve of Eratosthenes. Although this solution is slower than the best ones here, it is easy to grasp and designed to be readable...

I used integers to store the results of the sieve. In binary format, an integer is a list of 0s and 1s, 0 at position i if i is not a prime, 1 if it may be a prime. The requisite infinity is a result of the fact that Python 3 integers are unbounded.

def primes():
    container, size = 1 << 2, 3 # we start with 0b100 (from right to left: 0 and 1 are not primes, 2 is
    last_prime = 1
    while True:
        prime = next((j for j in range(last_prime+1, size) if container & 1 << j), None) # find the next prime
        while not prime:
            container, size = expand(container, size, 2**16) # add 65536 cells and sieve the container
            prime = next((j for j in range(last_prime+1, size) if container & 1 << j), None)
        yield prime
        last_prime = prime

How to expand the container? Just add a bunch of 1s at the left of the container (in binary format) and sieve them. This is identical to the standard sieve, with a slight difference. In the standard sieve, if we find a prime i, we start to cross the cells at i*i, with a step of i.

Here, this may have been done for the first part of container. We just need to start at the beginning of the new part of the container if it is farther than i*i.

def expand(container, size, n):
    new_size = size + n
    container += (1 << (new_size + 1) - 1) - (1 << size) # add n 1's
    for i in range(2, new_size):
        if container & (1 << i): # i is a prime
            t = sum(1 << j for j in range(max(i, size // i)*i, new_size, i)) # set 1 for all mutiple
            container &= ~t # cross the cells

    return container, new_size

Test for a million primes:

import itertools
assert 78498 == len(list(itertools.takewhile(lambda p: p<1000000, primes())))

Answer 3

For posterity, here's a rewrite of Will Ness's beautiful algorithm for Python 3. Some changes are needed (iterators no longer have .next() methods, but there's a new next() builtin function). Other changes are for fun (using the new yield from <iterable> replaces four yield statements in the original. More are for readability (I'm not a fan of overusing ;-) 1-letter variable names).

It's significantly faster than the original, but not for algorithmic reasons. The speedup is mostly due to removing the original's add() function, doing that inline instead.

def psieve():
    import itertools
    yield from (2, 3, 5, 7)
    D = {}
    ps = psieve()
    next(ps)
    p = next(ps)
    assert p == 3
    psq = p*p
    for i in itertools.count(9, 2):
        if i in D:      # composite
            step = D.pop(i)
        elif i < psq:   # prime
            yield i
            continue
        else:           # composite, = p*p
            assert i == psq
            step = 2*p
            p = next(ps)
            psq = p*p
        i += step
        while i in D:
            i += step
        D[i] = step

Answer 4

Here's a pretty fast infinite generator, written in Python2 but easily adjusted to Python3. To use it to add the primes up to 10**9, use the following:

from itertools import takewhile
from functools import partial
from operator import gt
print (sum(takewhile(partial(gt, 10**9), prime_gen_inf())))

It's a segmented sieve, faster but obviously less elegant than Will Ness's algorithm.

from operator import mul
from functools import reduce
def prod(x): return reduce(mul, x, 1)


def build_sieve(wheel):
    w = prod(wheel)
    w_phi = prod([p-1 for p in wheel])
    rems = [a for a in range(w) if all(a % p for p in wheel)]
    assert len(rems) == w_phi
    inv = {a:pow(a, w_phi - 1, w) for a in rems}
    try:
        known_p = wheel + rems[1 : rems.index(rems[1]*rems[1])]
    except ValueError:
        known_p = wheel + rems[1:]
    return wheel, w, w_phi, rems, inv, known_p

#Adjust the chunk variable based on your computer's architecture.
#
#Adjust the line with #! if you don't need "true" infinite.  If you don't need
#primes larger than 1<<32, use array('H', []), if 1<<64 use 'L', if 1<<128 (in
#Python3) use 'Q', otherwise use empty list [].
#To save memory, comment out the lines with #*, and uncomment the commented-out
#lines 
import itertools
from itertools import islice, count, compress, izip
chain_f = itertools.chain.from_iterable
from array import array
def prime_gen_inf(chunk=250000, sieve_info = build_sieve([2,3,5,7])):
    """    Indefinitely yields primes    """
    wheel, w, w_phi, rems, inv, known_p = sieve_info
    for p in known_p: yield p
    new_n = 0;
    while True:
        size = min(chunk, (p * p - new_n) / w)
        sieve = bytearray([1]) * size * w_phi
        n, new_n = new_n, new_n + size * w
        if not n:
            zero = bytearray([0])
            seen = len(known_p) - len(wheel) + 1
            sieve[:seen:1] = zero * seen
            p_gen = islice(prime_gen_inf(), len(wheel), None)
            new_p = next(p_gen)
            ps = []                                         #! array('H', [])
            p_invs = bytearray([])                                         #*
        while new_p * new_p < new_n:
            ps.append(new_p)
            p_invs.append(inv[new_p % w])                                  #*
            new_p = next(p_gen)
        for p, p_inv, modp in izip(ps, p_invs, [-n % p for p in ps]):      #*
            s = [(modp + p * (p_inv * (r - modp) % w)) / w for r in rems]  #*
        #for p in ps:
        #    s = [(-n%p + p * (inv[p%w] * (r - -n%p) % w)) / w for r in rems]
            for i, start in enumerate(s):
                slice_size = ((size - start - 1) / p + 1)
                sieve[i + start * w_phi :: p * w_phi] = zero * slice_size
        for p in compress(chain_f(izip(*[count(n+r, w) for r in rems])), sieve):
            yield p

Answer 5

Here is a simple but not terribly slow one using a heap instead of a dict:

import heapq

def heap_prime_gen_squares(): 
    yield 2  
    yield 3  
    h = [(9, 6)]
    n = 5
    while True:
        a, b = h[0]
        while n < a:
            yield n
            heapq.heappush(h, (n * n, n << 1))
            n += 2
        heapq.heapreplace(h, (a + b, b))  # Replace h[0], which is still (a, b).

My speed measurements of user time for the first 1 million primes (smaller numbers are better):

• postponed_sieve (dict-based): 8.553s
• erat2b (dict-based): 9.513s
• erat2a (dict-based): 10.313s
• heap_prime_gen_smallmem (heap-based): 23.935s
• heap_prime_gen_squares (heap-based): 27.302s
• heapprimegen (dict-based): 145.029s

So dict-based approaches seem to be the fastest.

Answer 6

Here is a complicated heap-based implementation, which is not much faster than other heap-based implementations (see the speed comparison in another answer of mine), but it uses much less memory.

This implementation uses two heaps (tu and wv), which contain the same number elements. Each element is an int pair. In order to find all primes up to q**2 (where q is a prime), each heap will contain at most 2*pi(q-1) elements, where pi(x) is the number of positive primes not larger than x. So the total number of integers is at most 4*pi(floor(sqrt(n))). (We could gain a factor on 2 on memory by pushing half as much stuff to the heap, but that would make the algorithm slower.)

Other dict and heap-based approaches (e.g. erat2b, and heap_prime_gen_squares and heapprimegen) above store about `2*pi(n)' integers, because they extend their heap or dict every time they find a prime. As a comparison: to find the 1_000_000 primes, this implementation stores less than 4141 integers, other implementations store more than 1_000_000 integers.

import heapq

def heap_prime_gen_smallmem():
    yield 2
    yield 3
    f = 5
    fmar3 = 2
    q = 7
    q6 = 7 * 6
    qmar3 = 4
    tu = [(25, 30), (35, 30)]
    vw = [(25, 30), (35, 30)]
    while True:
        qmar3 += 2   
        if qmar3 == 6:  
            qb = q + 4
            q6b = q6 + 24
            qmar3 = 2
        else:
            qb = q + 2
            q6b = q6 + 12
        if q < tu[0][0]:
            d = q * q
            while f < d:
                a, b = vw[0]
                if f < a: 
                    yield f   
                else:
                    a, b = vw[0]
                    heapq.heapreplace(vw, (a + b, b))
                    a, b = vw[0]
                    while f >= a:
                        heapq.heapreplace(vw, (a + b, b))
                        a, b = vw[0]   
                fmar3 += 2
                if fmar3 == 6:
                    f += 4
                    fmar3 = 2
                else:
                    f += 2
            c = q * qb   
            heapq.heappush(tu, (d, q6))
            heapq.heappush(tu, (c, q6))
            heapq.heappush(vw, (d, q6))
            heapq.heappush(vw, (c, q6))
        else:
            a, b = tu[0]
            heapq.heapreplace(tu, (a + b, b))
            a, b = tu[0]  
            while q >= a:
                heapq.heapreplace(tu, (a + b, b))
                a, b = tu[0]
        q = qb
        q6 = q6b

Answer 7

And another answer, more memory-efficient than my erat3 answer here:

import heapq

def heapprimegen():
    hp= []
    yield 2
    yield 3
    cn= 3
    nn, inc= 3, 6
    while 1:
        while cn < nn:
            yield cn
            heapq.heappush(hp, (3*cn, 2*cn))
            cn+= 2
        cn= nn+2
        nn, inc= heapq.heappushpop(hp, (nn+inc, inc))

It maintains a heap (a list) of prime multiples rather than a dictionary. It loses some speed, obviously.

Answer 8

Do a segmented sieve, where the size of a segment is determined by available memory or the maximal size of a bitset.

For each segment represent the numbers in some interval [n; n + segment_size) as a bit set and sieve with all prime numbers below the square root of the upper bound.

Using a bit set uses less memory than a hash table or tree data structure, because you are working with dense sets of numbers.

Answer 9

Another way to do it:

import itertools
def primeseq():
    prime = [2]
    num = 0
    yield 2
    for i in itertools.count(3, 2):
        is_prime = True
        for num in prime:
            if i % num == 0:
                is_prime = False
                break
            elif num ** 2 > i: 
                break
        if is_prime:
            prime.append(i)
            yield i

Answer 10

“If I have seen further…”

The erat2 function from the cookbook can be further sped up (by about 20-25%):

erat2a

import itertools as it
def erat2a( ):
    D = {  }
    yield 2
    for q in it.islice(it.count(3), 0, None, 2):
        p = D.pop(q, None)
        if p is None:
            D[q*q] = q
            yield q
        else:
            # old code here:
            # x = p + q
            # while x in D or not (x&1):
            #     x += p
            # changed into:
            x = q + 2*p
            while x in D:
                x += 2*p
            D[x] = p

The not (x&1) check verifies that x is odd. However, since both q and p are odd, by adding 2*p half of the steps are avoided along with the test for oddity.

erat3

If one doesn't mind a little extra fanciness, erat2 can be sped up by 35-40% with the following changes (NB: needs Python 2.7+ or Python 3+ because of the itertools.compress function):

import itertools as it
def erat3( ):
    D = { 9: 3, 25: 5 }
    yield 2
    yield 3
    yield 5
    MASK= 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0,
    MODULOS= frozenset( (1, 7, 11, 13, 17, 19, 23, 29) )

    for q in it.compress(
            it.islice(it.count(7), 0, None, 2),
            it.cycle(MASK)):
        p = D.pop(q, None)
        if p is None:
            D[q*q] = q
            yield q
        else:
            x = q + 2*p
            while x in D or (x%30) not in MODULOS:
                x += 2*p
            D[x] = p

The erat3 function takes advantage of the fact that all primes (except for 2, 3, 5) modulo 30 result to only eight numbers: the ones included in the MODULOS frozenset. Thus, after yielding the initial three primes, we start from 7 and work only with the candidates.
The candidate filtering uses the itertools.compress function; the “magic” is in the MASK sequence; MASK has 15 elements (there are 15 odd numbers in every 30 numbers, as chosen by the itertools.islice function) with a 1 for every possible candidate, starting from 7. The cycle repeats as specified by the itertools.cycle function.
The introduction of the candidate filtering needs another modification: the or (x%30) not in MODULOS check. The erat2 algorithm processed all odd numbers; now that the erat3 algorithm processes only r30 candidates, we need to make sure that all D.keys() can only be such —false— candidates.

Benchmarks

Results

On an Atom 330 Ubuntu 9.10 server, versions 2.6.4 and 3.1.1+:

$ testit
up to 8192
==== python2 erat2 ====
100 loops, best of 3: 18.6 msec per loop
==== python2 erat2a ====
100 loops, best of 3: 14.5 msec per loop
==== python2 erat3 ====
Traceback (most recent call last):
…
AttributeError: 'module' object has no attribute 'compress'
==== python3 erat2 ====
100 loops, best of 3: 19.2 msec per loop
==== python3 erat2a ====
100 loops, best of 3: 14.1 msec per loop
==== python3 erat3 ====
100 loops, best of 3: 11.7 msec per loop

On an AMD Geode LX Gentoo home server, Python 2.6.5 and 3.1.2:

$ testit
up to 8192
==== python2 erat2 ====
10 loops, best of 3: 104 msec per loop
==== python2 erat2a ====
10 loops, best of 3: 81 msec per loop
==== python2 erat3 ====
Traceback (most recent call last):
…
AttributeError: 'module' object has no attribute 'compress'
==== python3 erat2 ====
10 loops, best of 3: 116 msec per loop
==== python3 erat2a ====
10 loops, best of 3: 82 msec per loop
==== python3 erat3 ====
10 loops, best of 3: 66 msec per loop

Benchmark code

A primegen.py module contains the erat2, erat2a and erat3 functions. Here follows the testing script:

#!/bin/sh
max_num=${1:-8192}
echo up to $max_num
for python_version in python2 python3
do
    for function in erat2 erat2a erat3
    do
        echo "==== $python_version $function ===="
        $python_version -O -m timeit -c \
        -s  "import itertools as it, functools as ft, operator as op, primegen; cmp= ft.partial(op.ge, $max_num)" \
            "next(it.dropwhile(cmp, primegen.$function()))"
    done
done

Answer 11

This isn't originally my code, however, it's worth posting. The original can be found here: http://code.activestate.com/recipes/117119/

def gen_primes():
  D = {}
  q = 2  # first integer to test for primality.

  while True:
    if q not in D:
      # not marked composite, must be prime  
      yield q 

      #first multiple of q not already marked
      D[q * q] = [q] 
    else:
      for p in D[q]:
        D.setdefault(p + q, []).append(p)
      # no longer need D[q], free memory
      del D[q]

    q += 1

It's a generator, so use it like any other.

primes = gen_primes()
for p in primes:
  print p

It takes 1.62s to generate and put into a set, 1 million primes, on my desktop.

Answer 12

Here's a generator that's a little truer to how it's done in Haskell: filtering against composites of known primes, then adding the remaining primes to the list.

def gen_primes():
    primes = []
    i = 2
    while True:
        prime = True
        for p in primes:
            if not (i % p):
                prime = False
                break
        if prime:
            yield i
            primes.append(i)
        i += 1