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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><div id="lc1">from itertools import repeat, izip
</div><div id="lc2">from math import ceil, sqrt
</div><div id="lc3">from operator import eq, mod, truediv
</div><div id="lc4">
</div><div id="lc5">def _candidate_primes(limit):
</div><div id="lc6"> sqrt_limit = int(ceil(sqrt(limit)))
</div><div id="lc7"> for x in xrange(1, sqrt_limit):
</div><div id="lc8"> for y in xrange(1, sqrt_limit):
</div><div id="lc9"> n = 4*(x**2) + y**2
</div><div id="lc10"> if n &lt;= limit and (eq(mod(n, 12), 1) or eq(mod(n, 12), 5)):
</div><div id="lc11"> yield n
</div><div id="lc12"> n = 3 * (x**2) + y**2
</div><div id="lc13"> if n &lt;= limit and eq(mod(n, 12), 7):
</div><div id="lc14"> yield n
</div><div id="lc15"> n = 3*(x**2) - y**2
</div><div id="lc16"> if n &lt;= limit and x &gt; y and eq(mod(n, 12), 11):
</div><div id="lc17"> yield n 
</div><div id="lc18">
</div><div id="lc19">def atkins_sieve(limit):
</div><div id="lc20"> isprime = dict(izip((xrange(5, limit)), repeat(False, limit - 5)))
</div><div id="lc21"> for x in _candidate_primes(limit):
</div><div id="lc22"> try:
</div><div id="lc23"> isprime[x] = not isprime[x]
</div><div id="lc24"> except IndexError:
</div><div id="lc25"> continue
</div><div id="lc26"> yield 2
</div><div id="lc27"> yield 3
</div><div id="lc28"> for key in sorted(isprime.keys()):
</div><div id="lc29"> if isprime[key]:
</div><div id="lc30"> yield key
</div><div id="lc31"> for x in range(key**2, limit, key**2):
</div><div id="lc32"> isprime[x] = False
</div></pre></div>
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[color=#008000][b]from[/b][/color] [color=#0000FF][b]itertools[/b][/color] [color=#008000][b]import[/b][/color] repeat, izip
[color=#008000][b]from[/b][/color] [color=#0000FF][b]math[/b][/color] [color=#008000][b]import[/b][/color] ceil, sqrt
[color=#008000][b]from[/b][/color] [color=#0000FF][b]operator[/b][/color] [color=#008000][b]import[/b][/color] eq, mod, truediv

[color=#008000][b]def[/b][/color] [color=#0000FF]_candidate_primes[/color](limit):
 sqrt_limit [color=#666666]=[/color] [color=#008000]int[/color](ceil(sqrt(limit)))
 [color=#008000][b]for[/b][/color] x [color=#AA22FF][b]in[/b][/color] [color=#008000]xrange[/color]([color=#666666]1[/color], sqrt_limit):
 [color=#008000][b]for[/b][/color] y [color=#AA22FF][b]in[/b][/color] [color=#008000]xrange[/color]([color=#666666]1[/color], sqrt_limit):
 n [color=#666666]=[/color] [color=#666666]4[/color][color=#666666]*[/color](x[color=#666666]**[/color][color=#666666]2[/color]) [color=#666666]+[/color] y[color=#666666]**[/color][color=#666666]2[/color]
 [color=#008000][b]if[/b][/color] n [color=#666666]<=[/color] limit [color=#AA22FF][b]and[/b][/color] (eq(mod(n, [color=#666666]12[/color]), [color=#666666]1[/color]) [color=#AA22FF][b]or[/b][/color] eq(mod(n, [color=#666666]12[/color]), [color=#666666]5[/color])):
 [color=#008000][b]yield[/b][/color] n
 n [color=#666666]=[/color] [color=#666666]3[/color] [color=#666666]*[/color] (x[color=#666666]**[/color][color=#666666]2[/color]) [color=#666666]+[/color] y[color=#666666]**[/color][color=#666666]2[/color]
 [color=#008000][b]if[/b][/color] n [color=#666666]<=[/color] limit [color=#AA22FF][b]and[/b][/color] eq(mod(n, [color=#666666]12[/color]), [color=#666666]7[/color]):
 [color=#008000][b]yield[/b][/color] n
 n [color=#666666]=[/color] [color=#666666]3[/color][color=#666666]*[/color](x[color=#666666]**[/color][color=#666666]2[/color]) [color=#666666]-[/color] y[color=#666666]**[/color][color=#666666]2[/color]
 [color=#008000][b]if[/b][/color] n [color=#666666]<=[/color] limit [color=#AA22FF][b]and[/b][/color] x [color=#666666]>[/color] y [color=#AA22FF][b]and[/b][/color] eq(mod(n, [color=#666666]12[/color]), [color=#666666]11[/color]):
 [color=#008000][b]yield[/b][/color] n

[color=#008000][b]def[/b][/color] [color=#0000FF]atkins_sieve[/color](limit):
    isprime [color=#666666]=[/color] [color=#008000]dict[/color](izip(([color=#008000]xrange[/color]([color=#666666]5[/color], limit)), repeat([color=#008000]False[/color], limit [color=#666666]-[/color] [color=#666666]5[/color])))
    [color=#008000][b]for[/b][/color] x [color=#AA22FF][b]in[/b][/color] _candidate_primes(limit):
        [color=#008000][b]try[/b][/color]:
            isprime[x] [color=#666666]=[/color] [color=#AA22FF][b]not[/b][/color] isprime[x]
        [color=#008000][b]except[/b][/color] [color=#D2413A][b]IndexError[/b][/color]:
            [color=#008000][b]continue[/b][/color]
    [color=#008000][b]yield[/b][/color] [color=#666666]2[/color]
    [color=#008000][b]yield[/b][/color] [color=#666666]3[/color]
    [color=#008000][b]for[/b][/color] key [color=#AA22FF][b]in[/b][/color] [color=#008000]sorted[/color](isprime[color=#666666].[/color]keys()):
        [color=#008000][b]if[/b][/color] isprime[key]:
            [color=#008000][b]yield[/b][/color] key
            [color=#008000][b]for[/b][/color] x [color=#AA22FF][b]in[/b][/color] [color=#008000]range[/color](key[color=#666666]**[/color][color=#666666]2[/color], limit, key[color=#666666]**[/color][color=#666666]2[/color]):
                isprime[x] [color=#666666]=[/color] [color=#008000]False[/color]

--------------------------------------
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<div class="highlight"><pre><div id="lc1">from itertools import repeat, izip
</div><div id="lc2">from math import ceil, sqrt
</div><div id="lc3">from operator import eq, mod, truediv
</div><div id="lc4">
</div><div id="lc5">def _candidate_primes(limit):
</div><div id="lc6"> sqrt_limit = int(ceil(sqrt(limit)))
</div><div id="lc7"> for x in xrange(1, sqrt_limit):
</div><div id="lc8"> for y in xrange(1, sqrt_limit):
</div><div id="lc9"> n = 4*(x**2) + y**2
</div><div id="lc10"> if n &lt;= limit and (eq(mod(n, 12), 1) or eq(mod(n, 12), 5)):
</div><div id="lc11"> yield n
</div><div id="lc12"> n = 3 * (x**2) + y**2
</div><div id="lc13"> if n &lt;= limit and eq(mod(n, 12), 7):
</div><div id="lc14"> yield n
</div><div id="lc15"> n = 3*(x**2) - y**2
</div><div id="lc16"> if n &lt;= limit and x &gt; y and eq(mod(n, 12), 11):
</div><div id="lc17"> yield n 
</div><div id="lc18">
</div><div id="lc19">def atkins_sieve(limit):
</div><div id="lc20"> isprime = dict(izip((xrange(5, limit)), repeat(False, limit - 5)))
</div><div id="lc21"> for x in _candidate_primes(limit):
</div><div id="lc22"> try:
</div><div id="lc23"> isprime[x] = not isprime[x]
</div><div id="lc24"> except IndexError:
</div><div id="lc25"> continue
</div><div id="lc26"> yield 2
</div><div id="lc27"> yield 3
</div><div id="lc28"> for key in sorted(isprime.keys()):
</div><div id="lc29"> if isprime[key]:
</div><div id="lc30"> yield key
</div><div id="lc31"> for x in range(key**2, limit, key**2):
</div><div id="lc32"> isprime[x] = False
</div></pre></div>
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<blockquote><code>from&nbsp;itertools&nbsp;import&nbsp;repeat,&nbsp;izip
from&nbsp;math&nbsp;import&nbsp;ceil,&nbsp;sqrt
from&nbsp;operator&nbsp;import&nbsp;eq,&nbsp;mod,&nbsp;truediv

def&nbsp;_candidate_primes(limit):
&nbsp;&nbsp;&nbsp;&nbsp;sqrt_limit&nbsp;=&nbsp;int(ceil(sqrt(limit)))
&nbsp;&nbsp;&nbsp;&nbsp;for&nbsp;x&nbsp;in&nbsp;xrange(1,&nbsp;sqrt_limit):
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;for&nbsp;y&nbsp;in&nbsp;xrange(1,&nbsp;sqrt_limit):
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;n&nbsp;=&nbsp;4*(x**2)&nbsp;+&nbsp;y**2
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;if&nbsp;n&nbsp;&lt;=&nbsp;limit&nbsp;and&nbsp;(eq(mod(n,&nbsp;12),&nbsp;1)&nbsp;or&nbsp;eq(mod(n,&nbsp;12),&nbsp;5)):
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;yield&nbsp;n
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;n&nbsp;=&nbsp;3&nbsp;*&nbsp;(x**2)&nbsp;+&nbsp;y**2
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;if&nbsp;n&nbsp;&lt;=&nbsp;limit&nbsp;and&nbsp;eq(mod(n,&nbsp;12),&nbsp;7):
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;yield&nbsp;n
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;n&nbsp;=&nbsp;3*(x**2)&nbsp;-&nbsp;y**2
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;if&nbsp;n&nbsp;&lt;=&nbsp;limit&nbsp;and&nbsp;x&nbsp;&gt;&nbsp;y&nbsp;and&nbsp;eq(mod(n,&nbsp;12),&nbsp;11):
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;yield&nbsp;n&nbsp;&nbsp;&nbsp;

def&nbsp;atkins_sieve(limit):
&nbsp;&nbsp;&nbsp;&nbsp;isprime&nbsp;=&nbsp;dict(izip((xrange(5,&nbsp;limit)),&nbsp;repeat(False,&nbsp;limit&nbsp;-&nbsp;5)))
&nbsp;&nbsp;&nbsp;&nbsp;for&nbsp;x&nbsp;in&nbsp;_candidate_primes(limit):
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;try:
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;isprime[x]&nbsp;=&nbsp;not&nbsp;isprime[x]
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;except&nbsp;IndexError:
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;continue
&nbsp;&nbsp;&nbsp;&nbsp;yield&nbsp;2
&nbsp;&nbsp;&nbsp;&nbsp;yield&nbsp;3
&nbsp;&nbsp;&nbsp;&nbsp;for&nbsp;key&nbsp;in&nbsp;sorted(isprime.keys()):
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;if&nbsp;isprime[key]:
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;yield&nbsp;key
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;for&nbsp;x&nbsp;in&nbsp;range(key**2,&nbsp;limit,&nbsp;key**2):
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;isprime[x]&nbsp;=&nbsp;False
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