这是使用存储列表查找质数的一种优雅而简单的解决方案。从4个变量开始，你只需要测试除数的奇数质数，你只需要测试你要测试的质数的一半(测试9,11,13是否能整除17没有意义)。它将先前存储的质数作为除数进行测试。

    # Program to calculate Primes
 primes = [1,3,5,7]
for n in range(9,100000,2):
    for x in range(1,(len(primes)/2)):
        if n % primes[x] == 0:
            break
    else:
        primes.append(n)
print primes

第一次使用python，所以我在这里使用的一些方法可能看起来有点麻烦。我只是直接将我的c++代码转换为python，这就是我所拥有的(尽管在python中有点慢)

#!/usr/bin/env python
import time

def GetPrimes(n):

    Sieve = [1 for x in xrange(n)]

    Done = False
    w = 3

    while not Done:

        for q in xrange (3, n, 2):
            Prod = w*q
            if Prod < n:
                Sieve[Prod] = 0
            else:
                break

        if w > (n/2):
            Done = True
        w += 2

    return Sieve



start = time.clock()

d = 10000000
Primes = GetPrimes(d)

count = 1 #This is for 2

for x in xrange (3, d, 2):
    if Primes[x]:
        count+=1

elapsed = (time.clock() - start)
print "\nFound", count, "primes in", elapsed, "seconds!\n"

pythonw Primes.py 在12.799119秒内找到664579个质数!

#!/usr/bin/env python
import time

def GetPrimes2(n):

    Sieve = [1 for x in xrange(n)]

    for q in xrange (3, n, 2):
        k = q
        for y in xrange(k*3, n, k*2):
            Sieve[y] = 0

    return Sieve



start = time.clock()

d = 10000000
Primes = GetPrimes2(d)

count = 1 #This is for 2

for x in xrange (3, d, 2):
    if Primes[x]:
        count+=1

elapsed = (time.clock() - start)
print "\nFound", count, "primes in", elapsed, "seconds!\n"

pythonw Primes2.py 在10.230172秒内找到664579个质数!

#!/usr/bin/env python
import time

def GetPrimes3(n):

    Sieve = [1 for x in xrange(n)]

    for q in xrange (3, n, 2):
        k = q
        for y in xrange(k*k, n, k << 1):
            Sieve[y] = 0

    return Sieve



start = time.clock()

d = 10000000
Primes = GetPrimes3(d)

count = 1 #This is for 2

for x in xrange (3, d, 2):
    if Primes[x]:
        count+=1

elapsed = (time.clock() - start)
print "\nFound", count, "primes in", elapsed, "seconds!\n"

python Primes2.py 在7.113776秒内找到664579个质数!

从2021年的答案开始，我还没有发现二进制数组方法对10亿以下的质数有利。

但我可以用几个技巧将质数从2加速到接近x2:

使用numexpr库将numpy表达式转换为分配较少的紧循环 取代np。有更快的选择 以某种方式操作筛选的前9个元素，因此不需要改变数组的形状

总之，在我的机器上，质数< 10亿的时间从25秒变成了14.5秒

import numexpr as ne
import numpy as np

def primesfrom2to_numexpr(n):
    # https://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188
    """ Input n>=24, Returns a array of primes, 2 <= p < n + a few over"""
    sieve = np.zeros((n // 3 + (n % 6 == 2))//4+1, dtype=np.int32)
    ne.evaluate('sieve + 0x01010101', out=sieve)
    sieve = sieve.view('int8')
    #sieve = np.ones(n // 3 + (n % 6 == 2), dtype=np.bool_)
    sieve[0] = 0
    for i in np.arange(int(n ** 0.5) // 3 + 1):
        if sieve[i]:
            k = 3 * i + 1 | 1
            sieve[((k * k) // 3)::2 * k] = 0
            sieve[(k * k + 4 * k - 2 * k * (i & 1)) // 3::2 * k] = 0
    sieve[[0,8]] = 1
    result = np.flatnonzero(sieve)
    ne.evaluate('result * 3 + 1 + result%2', out=result)
    result[:9] = [2,3,5,7,11,13,17,19,23]
    return result

如果你不想重新发明轮子，你可以安装符号数学库symphony(是的，它与Python 3兼容)

pip install sympy

然后使用质数函数

from sympy import sieve
primes = list(sieve.primerange(1, 10**6))

你有一个更快的代码和最简单的代码生成质数。 但对于更大的数字，当n=10000, 10000000时，它不起作用，可能是。pop()方法失败了

考虑:N是质数吗?

case 1: You got some factors of N, for i in range(2, N): If N is prime loop is performed for ~(N-2) times. else less number of times case 2: for i in range(2, int(math.sqrt(N)): Loop is performed for almost ~(sqrt(N)-2) times if N is prime else will break somewhere case 3: Better We Divide N With Only number of primes<=sqrt(N) Where loop is performed for only π(sqrt(N)) times π(sqrt(N)) << sqrt(N) as N increases from math import sqrt from time import * prime_list = [2] n = int(input()) s = time() for n0 in range(2,n+1): for i0 in prime_list: if n0%i0==0: break elif i0>=int(sqrt(n0)): prime_list.append(n0) break e = time() print(e-s) #print(prime_list); print(f'pi({n})={len(prime_list)}') print(f'{n}: {len(prime_list)}, time: {e-s}') Output 100: 25, time: 0.00010275840759277344 1000: 168, time: 0.0008606910705566406 10000: 1229, time: 0.015588521957397461 100000: 9592, time: 0.023436546325683594 1000000: 78498, time: 4.1965954303741455 10000000: 664579, time: 109.24591708183289 100000000: 5761455, time: 2289.130858898163

小于1000似乎很慢，但小于10^6我认为更快。

然而，我无法理解时间的复杂性。

我测试了一些unutbu的功能，我用饥饿的百万数字计算它

获胜者是使用numpy库的函数，

注意:做一个内存利用率测试也很有趣:)

示例代码

完整的代码在我的github存储库

#!/usr/bin/env python

import lib
import timeit
import sys
import math
import datetime

import prettyplotlib as ppl
import numpy as np

import matplotlib.pyplot as plt
from prettyplotlib import brewer2mpl

primenumbers_gen = [
    'sieveOfEratosthenes',
    'ambi_sieve',
    'ambi_sieve_plain',
    'sundaram3',
    'sieve_wheel_30',
    'primesfrom3to',
    'primesfrom2to',
    'rwh_primes',
    'rwh_primes1',
    'rwh_primes2',
]

def human_format(num):
    # https://stackoverflow.com/questions/579310/formatting-long-numbers-as-strings-in-python?answertab=active#tab-top
    magnitude = 0
    while abs(num) >= 1000:
        magnitude += 1
        num /= 1000.0
    # add more suffixes if you need them
    return '%.2f%s' % (num, ['', 'K', 'M', 'G', 'T', 'P'][magnitude])


if __name__=='__main__':

    # Vars
    n = 10000000 # number itereration generator
    nbcol = 5 # For decompose prime number generator
    nb_benchloop = 3 # Eliminate false positive value during the test (bench average time)
    datetimeformat = '%Y-%m-%d %H:%M:%S.%f'
    config = 'from __main__ import n; import lib'
    primenumbers_gen = {
        'sieveOfEratosthenes': {'color': 'b'},
        'ambi_sieve': {'color': 'b'},
        'ambi_sieve_plain': {'color': 'b'},
         'sundaram3': {'color': 'b'},
        'sieve_wheel_30': {'color': 'b'},
# # #        'primesfrom2to': {'color': 'b'},
        'primesfrom3to': {'color': 'b'},
        # 'rwh_primes': {'color': 'b'},
        # 'rwh_primes1': {'color': 'b'},
        'rwh_primes2': {'color': 'b'},
    }


    # Get n in command line
    if len(sys.argv)>1:
        n = int(sys.argv[1])

    step = int(math.ceil(n / float(nbcol)))
    nbs = np.array([i * step for i in range(1, int(nbcol) + 1)])
    set2 = brewer2mpl.get_map('Paired', 'qualitative', 12).mpl_colors

    print datetime.datetime.now().strftime(datetimeformat)
    print("Compute prime number to %(n)s" % locals())
    print("")

    results = dict()
    for pgen in primenumbers_gen:
        results[pgen] = dict()
        benchtimes = list()
        for n in nbs:
            t = timeit.Timer("lib.%(pgen)s(n)" % locals(), setup=config)
            execute_times = t.repeat(repeat=nb_benchloop,number=1)
            benchtime = np.mean(execute_times)
            benchtimes.append(benchtime)
        results[pgen] = {'benchtimes':np.array(benchtimes)}

fig, ax = plt.subplots(1)
plt.ylabel('Computation time (in second)')
plt.xlabel('Numbers computed')
i = 0
for pgen in primenumbers_gen:

    bench = results[pgen]['benchtimes']
    avgs = np.divide(bench,nbs)
    avg = np.average(bench, weights=nbs)

    # Compute linear regression
    A = np.vstack([nbs, np.ones(len(nbs))]).T
    a, b = np.linalg.lstsq(A, nbs*avgs)[0]

    # Plot
    i += 1
    #label="%(pgen)s" % locals()
    #ppl.plot(nbs, nbs*avgs, label=label, lw=1, linestyle='--', color=set2[i % 12])
    label="%(pgen)s avg" % locals()
    ppl.plot(nbs, a * nbs + b, label=label, lw=2, color=set2[i % 12])
print datetime.datetime.now().strftime(datetimeformat)

ppl.legend(ax, loc='upper left', ncol=4)

# Change x axis label
ax.get_xaxis().get_major_formatter().set_scientific(False)
fig.canvas.draw()
labels = [human_format(int(item.get_text())) for item in ax.get_xticklabels()]

ax.set_xticklabels(labels)
ax = plt.gca()

plt.show()