这是我能想到的最好的算法。
def get_primes(n): numbers = set(range(n, 1, -1)) primes = [] while numbers: p = numbers.pop() primes.append(p) numbers.difference_update(set(range(p*2, n+1, p))) return primes >>> timeit.Timer(stmt='get_primes.get_primes(1000000)', setup='import get_primes').timeit(1) 1.1499958793645562
可以使它更快吗？
此代码有一个缺陷：由于numbers是无序集合，因此不能保证numbers.pop()从集合中删除最低的数字。但是，它对某些输入数字有效（至少对我而言）：
numbers
numbers.pop()
>>> sum(get_primes(2000000)) 142913828922L #That's the correct sum of all numbers below 2 million >>> 529 in get_primes(1000) False >>> 529 in get_primes(530) True
警告： timeit由于硬件或Python版本的差异，结果可能会有所不同。
timeit
下面是一个脚本，它比较了许多实现：
在使用psyco测试的简单Python方法中，对于n = 1000000， rwh_primes1是测试最快的方法。
rwh_primes1
+---------------------+-------+ | Method | ms | +---------------------+-------+ | rwh_primes1 | 43.0 | | sieveOfAtkin | 46.4 | | rwh_primes | 57.4 | | sieve_wheel_30 | 63.0 | | rwh_primes2 | 67.8 | | sieveOfEratosthenes | 147.0 | | ambi_sieve_plain | 152.0 | | sundaram3 | 194.0 | +---------------------+-------+
在没有psyco的情况下，经过测试的普通Python方法中n = 1000000， rwh_primes2是最快的。
+---------------------+-------+ | Method | ms | +---------------------+-------+ | rwh_primes2 | 68.1 | | rwh_primes1 | 93.7 | | rwh_primes | 94.6 | | sieve_wheel_30 | 97.4 | | sieveOfEratosthenes | 178.0 | | ambi_sieve_plain | 286.0 | | sieveOfAtkin | 314.0 | | sundaram3 | 416.0 | +---------------------+-------+
在所有测试的方法中，允许numpy，对于n = 1000000， primesfrom2to是测试最快的方法。
+---------------------+-------+ | Method | ms | +---------------------+-------+ | primesfrom2to | 15.9 | | primesfrom3to | 18.4 | | ambi_sieve | 29.3 | +---------------------+-------+
使用以下命令测量时间：
python -mtimeit -s"import primes" "primes.{method}(1000000)"
用{method}每个方法名称替换。
{method}
primes.py： #!/usr/bin/env python import psyco; psyco.full() from math import sqrt, ceil import numpy as np def rwh_primes(n): # https://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188 """ Returns a list of primes < n """ sieve = [True] * n for i in xrange(3,int(n**0.5)+1,2): if sieve[i]: sieve[i*i::2*i]=[False]*((n-i*i-1)/(2*i)+1) return [2] + [i for i in xrange(3,n,2) if sieve[i]] def rwh_primes1(n): # https://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188 """ Returns a list of primes < n """ sieve = [True] * (n/2) for i in xrange(3,int(n**0.5)+1,2): if sieve[i/2]: sieve[i*i/2::i] = [False] * ((n-i*i-1)/(2*i)+1) return [2] + [2*i+1 for i in xrange(1,n/2) if sieve[i]] def rwh_primes2(n): # https://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188 """ Input n>=6, Returns a list of primes, 2 <= p < 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 xrange(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 xrange(1,n/3-correction) if sieve[i]] def sieve_wheel_30(N): # http://zerovolt.com/?p=88 ''' Returns a list of primes <= N using wheel criterion 2*3*5 = 30 Copyright 2009 by zerovolt.com This code is free for non-commercial purposes, in which case you can just leave this comment as a credit for my work. If you need this code for commercial purposes, please contact me by sending an email to: info [at] zerovolt [dot] com.''' __smallp = ( 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, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997) wheel = (2, 3, 5) const = 30 if N < 2: return [] if N <= const: pos = 0 while __smallp[pos] <= N: pos += 1 return list(__smallp[:pos]) # make the offsets list offsets = (7, 11, 13, 17, 19, 23, 29, 1) # prepare the list p = [2, 3, 5] dim = 2 + N // const tk1 = [True] * dim tk7 = [True] * dim tk11 = [True] * dim tk13 = [True] * dim tk17 = [True] * dim tk19 = [True] * dim tk23 = [True] * dim tk29 = [True] * dim tk1[0] = False # help dictionary d # d[a , b] = c ==> if I want to find the smallest useful multiple of (30*pos)+a # on tkc, then I need the index given by the product of [(30*pos)+a][(30*pos)+b] # in general. If b < a, I need [(30*pos)+a][(30*(pos+1))+b] d = {} for x in offsets: for y in offsets: res = (x*y) % const if res in offsets: d[(x, res)] = y # another help dictionary: gives tkx calling tmptk[x] tmptk = {1:tk1, 7:tk7, 11:tk11, 13:tk13, 17:tk17, 19:tk19, 23:tk23, 29:tk29} pos, prime, lastadded, stop = 0, 0, 0, int(ceil(sqrt(N))) # inner functions definition def del_mult(tk, start, step): for k in xrange(start, len(tk), step): tk[k] = False # end of inner functions definition cpos = const * pos while prime < stop: # 30k + 7 if tk7[pos]: prime = cpos + 7 p.append(prime) lastadded = 7 for off in offsets: tmp = d[(7, off)] start = (pos + prime) if off == 7 else (prime * (const * (pos + 1 if tmp < 7 else 0) + tmp) )//const del_mult(tmptk[off], start, prime) # 30k + 11 if tk11[pos]: prime = cpos + 11 p.append(prime) lastadded = 11 for off in offsets: tmp = d[(11, off)] start = (pos + prime) if off == 11 else (prime * (const * (pos + 1 if tmp < 11 else 0) + tmp) )//const del_mult(tmptk[off], start, prime) # 30k + 13 if tk13[pos]: prime = cpos + 13 p.append(prime) lastadded = 13 for off in offsets: tmp = d[(13, off)] start = (pos + prime) if off == 13 else (prime * (const * (pos + 1 if tmp < 13 else 0) + tmp) )//const del_mult(tmptk[off], start, prime) # 30k + 17 if tk17[pos]: prime = cpos + 17 p.append(prime) lastadded = 17 for off in offsets: tmp = d[(17, off)] start = (pos + prime) if off == 17 else (prime * (const * (pos + 1 if tmp < 17 else 0) + tmp) )//const del_mult(tmptk[off], start, prime) # 30k + 19 if tk19[pos]: prime = cpos + 19 p.append(prime) lastadded = 19 for off in offsets: tmp = d[(19, off)] start = (pos + prime) if off == 19 else (prime * (const * (pos + 1 if tmp < 19 else 0) + tmp) )//const del_mult(tmptk[off], start, prime) # 30k + 23 if tk23[pos]: prime = cpos + 23 p.append(prime) lastadded = 23 for off in offsets: tmp = d[(23, off)] start = (pos + prime) if off == 23 else (prime * (const * (pos + 1 if tmp < 23 else 0) + tmp) )//const del_mult(tmptk[off], start, prime) # 30k + 29 if tk29[pos]: prime = cpos + 29 p.append(prime) lastadded = 29 for off in offsets: tmp = d[(29, off)] start = (pos + prime) if off == 29 else (prime * (const * (pos + 1 if tmp < 29 else 0) + tmp) )//const del_mult(tmptk[off], start, prime) # now we go back to top tk1, so we need to increase pos by 1 pos += 1 cpos = const * pos # 30k + 1 if tk1[pos]: prime = cpos + 1 p.append(prime) lastadded = 1 for off in offsets: tmp = d[(1, off)] start = (pos + prime) if off == 1 else (prime * (const * pos + tmp) )//const del_mult(tmptk[off], start, prime) # time to add remaining primes # if lastadded == 1, remove last element and start adding them from tk1 # this way we don't need an "if" within the last while if lastadded == 1: p.pop() # now complete for every other possible prime while pos < len(tk1): cpos = const * pos if tk1[pos]: p.append(cpos + 1) if tk7[pos]: p.append(cpos + 7) if tk11[pos]: p.append(cpos + 11) if tk13[pos]: p.append(cpos + 13) if tk17[pos]: p.append(cpos + 17) if tk19[pos]: p.append(cpos + 19) if tk23[pos]: p.append(cpos + 23) if tk29[pos]: p.append(cpos + 29) pos += 1 # remove exceeding if present pos = len(p) - 1 while p[pos] > N: pos -= 1 if pos < len(p) - 1: del p[pos+1:] # return p list return p def sieveOfEratosthenes(n): """sieveOfEratosthenes(n): return the list of the primes < n.""" # Code from: <dickinsm@gmail.com>, Nov 30 2006 # http://groups.google.com/group/comp.lang.python/msg/f1f10ced88c68c2d if n <= 2: return [] sieve = range(3, n, 2) top = len(sieve) for si in sieve: if si: bottom = (si*si - 3) // 2 if bottom >= top: break sieve[bottom::si] = [0] * -((bottom - top) // si) return [2] + [el for el in sieve if el] def sieveOfAtkin(end): """sieveOfAtkin(end): return a list of all the prime numbers <end using the Sieve of Atkin.""" # Code by Steve Krenzel, <Sgk284@gmail.com>, improved # Code: https://web.archive.org/web/20080324064651/http://krenzel.info/?p=83 # Info: http://en.wikipedia.org/wiki/Sieve_of_Atkin assert end > 0 lng = ((end-1) // 2) sieve = [False] * (lng + 1) x_max, x2, xd = int(sqrt((end-1)/4.0)), 0, 4 for xd in xrange(4, 8*x_max + 2, 8): x2 += xd y_max = int(sqrt(end-x2)) n, n_diff = x2 + y_max*y_max, (y_max << 1) - 1 if not (n & 1): n -= n_diff n_diff -= 2 for d in xrange((n_diff - 1) << 1, -1, -8): m = n % 12 if m == 1 or m == 5: m = n >> 1 sieve[m] = not sieve[m] n -= d x_max, x2, xd = int(sqrt((end-1) / 3.0)), 0, 3 for xd in xrange(3, 6 * x_max + 2, 6): x2 += xd y_max = int(sqrt(end-x2)) n, n_diff = x2 + y_max*y_max, (y_max << 1) - 1 if not(n & 1): n -= n_diff n_diff -= 2 for d in xrange((n_diff - 1) << 1, -1, -8): if n % 12 == 7: m = n >> 1 sieve[m] = not sieve[m] n -= d x_max, y_min, x2, xd = int((2 + sqrt(4-8*(1-end)))/4), -1, 0, 3 for x in xrange(1, x_max + 1): x2 += xd xd += 6 if x2 >= end: y_min = (((int(ceil(sqrt(x2 - end))) - 1) << 1) - 2) << 1 n, n_diff = ((x*x + x) << 1) - 1, (((x-1) << 1) - 2) << 1 for d in xrange(n_diff, y_min, -8): if n % 12 == 11: m = n >> 1 sieve[m] = not sieve[m] n += d primes = [2, 3] if end <= 3: return primes[:max(0,end-2)] for n in xrange(5 >> 1, (int(sqrt(end))+1) >> 1): if sieve[n]: primes.append((n << 1) + 1) aux = (n << 1) + 1 aux *= aux for k in xrange(aux, end, 2 * aux): sieve[k >> 1] = False s = int(sqrt(end)) + 1 if s % 2 == 0: s += 1 primes.extend([i for i in xrange(s, end, 2) if sieve[i >> 1]]) return primes def ambi_sieve_plain(n): s = range(3, n, 2) for m in xrange(3, int(n**0.5)+1, 2): if s[(m-3)/2]: for t in xrange((m*m-3)/2,(n>>1)-1,m): s[t]=0 return [2]+[t for t in s if t>0] def sundaram3(max_n): # https://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/2073279#2073279 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) ################################################################################ # Using Numpy: def ambi_sieve(n): # http://tommih.blogspot.com/2009/04/fast-prime-number-generator.html s = np.arange(3, n, 2) for m in xrange(3, int(n ** 0.5)+1, 2): if s[(m-3)/2]: s[(m*m-3)/2::m]=0 return np.r_[2, s[s>0]] def primesfrom3to(n): # https://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188 """ Returns a array of primes, p < n """ assert n>=2 sieve = np.ones(n/2, dtype=np.bool) for i in xrange(3,int(n**0.5)+1,2): if sieve[i/2]: sieve[i*i/2::i] = False return np.r_[2, 2*np.nonzero(sieve)[0][1::]+1] 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 xrange(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)] if __name__=='__main__': import itertools import sys def test(f1,f2,num): print('Testing {f1} and {f2} return same results'.format( f1=f1.func_name, f2=f2.func_name)) if not all([a==b for a,b in itertools.izip_longest(f1(num),f2(num))]): sys.exit("Error: %s(%s) != %s(%s)"%(f1.func_name,num,f2.func_name,num)) n=1000000 test(sieveOfAtkin,sieveOfEratosthenes,n) test(sieveOfAtkin,ambi_sieve,n) test(sieveOfAtkin,ambi_sieve_plain,n) test(sieveOfAtkin,sundaram3,n) test(sieveOfAtkin,sieve_wheel_30,n) test(sieveOfAtkin,primesfrom3to,n) test(sieveOfAtkin,primesfrom2to,n) test(sieveOfAtkin,rwh_primes,n) test(sieveOfAtkin,rwh_primes1,n) test(sieveOfAtkin,rwh_primes2,n)
运行脚本会测试所有实现都给出相同的结果。
