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    <p>The first thing to note is that what you have written is not the sieve of eratosthenes. Look how many loops a totally naive sieve of eratosthenes executes:</p> <pre><code>def sieve1(n): loops = 0 numbers = set(range(2, n)) for i in range(2, int(n ** 0.5) + 1): for j in range(i * 2, n, i): numbers.discard(j) loops += 1 return sorted(numbers), loops </code></pre> <p>Tested:</p> <pre><code>&gt;&gt;&gt; sieve1(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], 178) </code></pre> <p>178 loops for 100 numbers (not including the sort). This can be improved with a few minor changes:</p> <pre><code>def sieve2(n): loops = 0 numbers = range(0, n) for prime in numbers: if prime &lt; 2: continue elif prime &gt; n ** 0.5: break for i in range(prime ** 2, n, prime): numbers[i] = 0 loops += 1 return [x for x in numbers if x &gt; 1], loops </code></pre> <p>Tested:</p> <pre><code>&gt;&gt;&gt; sieve2(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], 102) </code></pre> <p>102 loops for 100 numbers (not including the filter at the end). Now look at yours:</p> <pre><code>def sieve3(n): loops = 0 numbers = range(2, n) for i in numbers: for j in numbers: if j != i and j % i == 0: numbers.remove(j) loops += 1 return numbers, loops </code></pre> <p>Tested:</p> <pre><code>&gt;&gt;&gt; sieve3(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], 663) </code></pre> <p>It gets worse:</p> <pre><code>&gt;&gt;&gt; [sieve1(x)[1] for x in [100, 1000, 10000]] [178, 2978, 41723] &gt;&gt;&gt; [sieve2(x)[1] for x in [100, 1000, 10000]] [102, 1409, 16979] &gt;&gt;&gt; [sieve3(x)[1] for x in [100, 1000, 10000]] [663, 28986, 1523699] </code></pre> <p>At <code>n = 10000</code>, your implementation does almost 100x as much work!</p> <p>My suggestion would be to create a sensible implementation before making it "compact." Code golf can be fun, but it's nowhere near as challenging or as edifying as writing <em>efficient</em> code, whatever the length. </p> <p>That said, consider this one-liner (if you don't count the import, which you could get rid of by using <code>lambda x, y: x - y</code> in place of <code>operator.sub</code>). This implements the first algorithm with a small improvement:</p> <pre><code>&gt;&gt;&gt; from operator import sub &gt;&gt;&gt; reduce(sub, (set(range(x ** 2, 100, x)) for x in range(2, int(100 ** 0.5) + 1)), set(range(2, 100))) set([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]) </code></pre>
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    1. POPython- Sieve of Eratosthenes- Compact Python
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    1. COThis is impressive and incredibly instructive. I'm a programming amateur, so I apologize for its inefficiency :-\
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    2. CO@Parseltongue, no need to apologize for that -- everyone writes inefficient code! And _my_ apologies if I was curt -- my concern isn't really about efficiency, but about what one should [prioritize as a programmer](http://c2.com/cgi/wiki?MakeItWorkMakeItRightMakeItFast). And to be fair, aiming for compactness is important sometimes; it's just that overcompressed code is rarely ideal.
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