Schnellster Weg, um alle Primzahlen unter N aufzulisten

Question

Dies ist der beste Algorithmus, den ich entwickeln konnte.

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

Kann er noch schneller gemacht werden?

Dieser Code hat ein Problem: Da numbers eine ungeordnete Menge ist, gibt es keine Garantie dafür, dass numbers.pop() die niedrigste Zahl aus der Menge entfernt. Trotzdem funktioniert es (zumindest für mich) für einige Eingabelnummern:

>>> sum(get_primes(2000000))
142913828922L
#Das ist die korrekte Summe aller Zahlen unter 2 Millionen
>>> 529 in get_primes(1000)
False
>>> 529 in get_primes(530)
True

Answer 1

Für die wirklich schnellste Lösung mit ausreichend großem N wäre es, eine vorab berechnete Liste von Primzahlen herunterzuladen, sie als Tupel zu speichern und etwas Ähnliches zu tun:

for pos,i in enumerate(primes):
    if i > N:
        print primes[:pos]

Wenn N > primes[-1] ist, dann berechne nur mehr Primzahlen und speichere die neue Liste in deinem Code, so dass es beim nächsten Mal genauso schnell ist.

Denk immer über den Tellerrand hinaus.

Answer 2

Wenn Sie das Rad nicht neu erfinden möchten, können Sie die symbolische Mathematik-Bibliothek sympy installieren (ja, es ist Python 3 kompatibel)

pip install sympy

Und verwenden Sie die primerange Funktion

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

Answer 3

Wenn Sie itertools akzeptieren, aber nicht numpy, hier ist eine Anpassung von rwh_primes2 für Python 3, die auf meinem Computer etwa doppelt so schnell läuft. Der einzige wesentliche Unterschied besteht darin, dass anstelle einer Liste ein bytearray für den Booleschen Wert verwendet wird und dass anstelle einer Listenabstraktion compress zur Erstellung der endgültigen Liste verwendet wird. (Ich würde dies wie moarningsun als Kommentar hinzufügen, wenn ich dazu in der Lage wäre.)

import itertools
izip = itertools.zip_longest
chain = itertools.chain.from_iterable
compress = itertools.compress
def rwh_primes2_python3(n):
    """ Eingabe n>=6, liefert eine Liste der Primzahlen, 2 <= p < n """
    zero = bytearray([False])
    size = n//3 + (n % 6 == 2)
    sieve = bytearray([True]) * size
    sieve[0] = False
    for i in range(int(n**0.5)//3+1):
      if sieve[i]:
        k=3*i+1|1
        start = (k*k+4*k-2*k*(i&1))//3
        sieve[(k*k)//3::2*k]=zero*((size - (k*k)//3 - 1) // (2 * k) + 1)
        sieve[  start ::2*k]=zero*((size -   start  - 1) // (2 * k) + 1)
    ans = [2,3]
    poss = chain(izip(*[range(i, n, 6) for i in (1,5)]))
    ans.extend(compress(poss, sieve))
    return ans

Vergleiche:

>>> timeit.timeit('primes.rwh_primes2(10**6)', setup='import primes', number=1)
0.0652179726976101
>>> timeit.timeit('primes.rwh_primes2_python3(10**6)', setup='import primes', number=1)
0.03267321276325674

und

>>> timeit.timeit('primes.rwh_primes2(10**8)', setup='import primes', number=1)
6.394284538007014
>>> timeit.timeit('primes.rwh_primes2_python3(10**8)', setup='import primes', number=1)
3.833829450302801

Answer 4

Ich habe einen großen Teil des Codes für Python 3 aktualisiert und ihn unter Perfplot (ein Projekt von mir), um herauszufinden, was tatsächlich am schnellsten ist. Es stellte sich heraus, dass für große n , primesfrom{2,3}to den Vogel ab:

---

Code zur Reproduktion des Plots:

import perfplot
from math import sqrt, ceil
import numpy as np
import sympy

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 range(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 range(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 range(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 range(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"""
    assert n >= 6
    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]]

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 range(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 sieve_of_eratosthenes(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 = list(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 sieve_of_atkin(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 range(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 range((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 range(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 range((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 range(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 range(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 range(5 >> 1, (int(sqrt(end)) + 1) >> 1):
        if sieve[n]:
            primes.append((n << 1) + 1)
            aux = (n << 1) + 1
            aux *= aux
            for k in range(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 range(s, end, 2) if sieve[i >> 1]])

    return primes

def ambi_sieve_plain(n):
    s = list(range(3, n, 2))
    for m in range(3, int(n ** 0.5) + 1, 2):
        if s[(m - 3) // 2]:
            for t in range((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 range(3, max_n + 1, 2):
        for i in range(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 range(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 an array of primes, p < n """
    assert n >= 2
    sieve = np.ones(n // 2, dtype=bool)
    for i in range(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 an array of primes, 2 <= p < n """
    assert n >= 6
    sieve = np.ones(n // 3 + (n % 6 == 2), dtype=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)]

def sympy_sieve(n):
    return list(sympy.sieve.primerange(1, n))

b = perfplot.bench(
    setup=lambda n: n,
    kernels=[
        rwh_primes,
        rwh_primes1,
        rwh_primes2,
        sieve_wheel_30,
        sieve_of_eratosthenes,
        sieve_of_atkin,
        # ambi_sieve_plain,
        # sundaram3,
        ambi_sieve,
        primesfrom3to,
        primesfrom2to,
        sympy_sieve,
    ],
    n_range=[2 ** k for k in range(3, 25)],
    xlabel="n",
)
b.save("out.png")
b.show()

Answer 5

Hier sind zwei aktualisierte (reine Python 3.6) Versionen einer der schnellsten Funktionen,

from itertools import compress

def rwh_primes1v1(n):
    """Gibt eine Liste von Primzahlen  2 zurück"""
    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)
    return [2,*compress(range(3,n,2), sieve[1:])]

def rwh_primes1v2(n):
    """Gibt eine Liste von Primzahlen  2 zurück"""
    sieve = bytearray([True]) * (n//2+1)
    for i in range(1,int(n**0.5)//2+1):
        if sieve[i]:
            sieve[2*i*(i+1)::2*i+1] = bytearray((n//2-2*i*(i+1))//(2*i+1)+1)
    return [2,*compress(range(3,n,2), sieve[1:])]