对于Python 3

def rwh_primes2(n):
    correction = (n%6>1)
    n = {0:n,1:n-1,2:n+4,3:n+3,4:n+2,5:n+1}[n%6]
    sieve = [True] * (n//3)
    sieve[0] = False
    for i in range(int(n**0.5)//3+1):
      if sieve[i]:
        k=3*i+1|1
        sieve[      ((k*k)//3)      ::2*k]=[False]*((n//6-(k*k)//6-1)//k+1)
        sieve[(k*k+4*k-2*k*(i&1))//3::2*k]=[False]*((n//6-(k*k+4*k-2*k*(i&1))//6-1)//k+1)
    return [2,3] + [3*i+1|1 for i in range(1,n//3-correction) if sieve[i]]

使用Sundaram的Sieve，我想我打破了pure-Python的记录:

def sundaram3(max_n):
    numbers = range(3, max_n+1, 2)
    half = (max_n)//2
    initial = 4

    for step in xrange(3, max_n+1, 2):
        for i in xrange(initial, half, step):
            numbers[i-1] = 0
        initial += 2*(step+1)

        if initial > half:
            return [2] + filter(None, numbers)

Comparasion:

C:\USERS>python -m timeit -n10 -s "import get_primes" "get_primes.get_primes_erat(1000000)"
10 loops, best of 3: 710 msec per loop

C:\USERS>python -m timeit -n10 -s "import get_primes" "get_primes.daniel_sieve_2(1000000)"
10 loops, best of 3: 435 msec per loop

C:\USERS>python -m timeit -n10 -s "import get_primes" "get_primes.sundaram3(1000000)"
10 loops, best of 3: 327 msec per loop

这是问题解的一种变化应该比问题本身更快。它使用埃拉托色尼的静态筛，没有其他优化。

from typing import List

def list_primes(limit: int) -> List[int]:
    primes = set(range(2, limit + 1))
    for i in range(2, limit + 1):
        if i in primes:
            primes.difference_update(set(list(range(i, limit + 1, i))[1:]))
    return sorted(primes)

>>> list_primes(100)
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]

在Pure Python中最快的质数筛分:

from itertools import compress

def half_sieve(n):
    """
    Returns a list of prime numbers less than `n`.
    """
    if n <= 2:
        return []
    sieve = bytearray([True]) * (n // 2)
    for i in range(3, int(n ** 0.5) + 1, 2):
        if sieve[i // 2]:
            sieve[i * i // 2::i] = bytearray((n - i * i - 1) // (2 * i) + 1)
    primes = list(compress(range(1, n, 2), sieve))
    primes[0] = 2
    return primes

我优化了埃拉托色尼筛子的速度和内存。

基准

from time import clock
import platform

def benchmark(iterations, limit):
    start = clock()
    for x in range(iterations):
        half_sieve(limit)
    end = clock() - start
    print(f'{end/iterations:.4f} seconds for primes < {limit}')

if __name__ == '__main__':
    print(platform.python_version())
    print(platform.platform())
    print(platform.processor())
    it = 10
    for pw in range(4, 9):
        benchmark(it, 10**pw)

输出

>>> 3.6.7
>>> Windows-10-10.0.17763-SP0
>>> Intel64 Family 6 Model 78 Stepping 3, GenuineIntel
>>> 0.0003 seconds for primes < 10000
>>> 0.0021 seconds for primes < 100000
>>> 0.0204 seconds for primes < 1000000
>>> 0.2389 seconds for primes < 10000000
>>> 2.6702 seconds for primes < 100000000

很抱歉打扰，但erat2()在算法中有一个严重的缺陷。

在搜索下一个合成时，我们只需要测试奇数。 Q p都是奇数;那么q+p是偶数，不需要检验，但q+2*p总是奇数。这消除了while循环条件中的“if even”测试，并节省了大约30%的运行时。

当我们在它:而不是优雅的'D.pop(q,None)'获取和删除方法，使用'if q in D: p=D[q]，del D[q]'，这是两倍的速度!至少在我的机器上(P3-1Ghz)。 所以我建议这个聪明算法的实现:

def erat3( ):
    from itertools import islice, count

    # q is the running integer that's checked for primeness.
    # yield 2 and no other even number thereafter
    yield 2
    D = {}
    # no need to mark D[4] as we will test odd numbers only
    for q in islice(count(3),0,None,2):
        if q in D:                  #  is composite
            p = D[q]
            del D[q]
            # q is composite. p=D[q] is the first prime that
            # divides it. Since we've reached q, we no longer
            # need it in the map, but we'll mark the next
            # multiple of its witnesses to prepare for larger
            # numbers.
            x = q + p+p        # next odd(!) multiple
            while x in D:      # skip composites
                x += p+p
            D[x] = p
        else:                  # is prime
            # q is a new prime.
            # Yield it and mark its first multiple that isn't
            # already marked in previous iterations.
            D[q*q] = q
            yield q

在写这篇文章的时候，这是最快的工作解决方案(至少在我的机器上是这样)。它同时使用numpy和bitarray，并受到这个答案的primesfrom2to的启发。

import numpy as np
from bitarray import bitarray


def bit_primes(n):
    bit_sieve = bitarray(n // 3 + (n % 6 == 2))
    bit_sieve.setall(1)
    bit_sieve[0] = False

    for i in range(int(n ** 0.5) // 3 + 1):
        if bit_sieve[i]:
            k = 3 * i + 1 | 1
            bit_sieve[k * k // 3::2 * k] = False
            bit_sieve[(k * k + 4 * k - 2 * k * (i & 1)) // 3::2 * k] = False

    np_sieve = np.unpackbits(np.frombuffer(bit_sieve.tobytes(), dtype=np.uint8)).view(bool)
    return np.concatenate(((2, 3), ((3 * np.flatnonzero(np_sieve) + 1) | 1)))

下面是与素数from2to的比较，它之前被发现是unutbu比较中最快的解:

python3 -m timeit -s "import fast_primes" "fast_primes.bit_primes(1000000)"
200 loops, best of 5: 1.19 msec per loop

python3 -m timeit -s "import fast_primes" "fast_primes.primesfrom2to(1000000)"
200 loops, best of 5: 1.23 msec per loop

对于寻找100万以下的质数，bit_primes稍微快一些。 n值越大，差异就越大。在某些情况下，bit_primes的速度是原来的两倍多:

python3 -m timeit -s "import fast_primes" "fast_primes.bit_primes(500_000_000)"
1 loop, best of 5: 540 msec per loop

python3 -m timeit -s "import fast_primes" "fast_primes.primesfrom2to(500_000_000)"
1 loop, best of 5: 1.15 sec per loop

作为参考，以下是primesfrom2to I的最小修改版本(适用于Python 3):

def primesfrom2to(n):
    # https://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188
    """ Input n>=6, Returns a array of primes, 2 <= p < n"""
    sieve = np.ones(n // 3 + (n % 6 == 2), dtype=np.bool)
    sieve[0] = False
    for i in range(int(n ** 0.5) // 3 + 1):
        if sieve[i]:
            k = 3 * i + 1 | 1
            sieve[((k * k) // 3)::2 * k] = False
            sieve[(k * k + 4 * k - 2 * k * (i & 1)) // 3::2 * k] = False
    return np.r_[2, 3, ((3 * np.nonzero(sieve)[0] + 1) | 1)]