We start off by trying to translate the description in the wikipedia article into python line by line. Which gives us the following.

def esieve(n): primes = range(2, n+1) p = 2 while p < n: for i in range(p, n+1): if p*i in primes: primes.remove(p*i) p += 1 return primes9 Lines isn't bad for a start, but that if statement can be cleaned up. What if instead of looking for items in our list one at a time. We make a new list of items to be removed, and remove them all at the end? We could use a set to hold our lists. This lets us use the minus (-) operator to give us a new set of items not in our marked set.

def shorter_esieve(n): marked = set() p = 2 while p < n: for i in range(p, n+1): marked.add(p*i) p += 1 return sorted(set(range(2, n+1)) - marked)We only removed one line in that last attempt. Not great. But it looks like we're using a while loop and incrementing each step. Why don't we just do a for?

def shorter_esieve(n): marked = set() for p in range(2, n+1): for i in range(p, n+1): marked.add(p*i) return sorted(set(range(2, n+1)) - marked)6 lines, getting better. Now here is the magic. We're using two for loops to generate a set of values. So we can just use a list comprehension to build our list, which we then use to make our marked set.

def shorter_esieve(n): marked = set([p* i for p in range(2, n+1) for i in range(p, n+1)]) return sorted(set(range(2, n+1)) - marked)And moving the assignment inline

def much_shorter_esieve(n): return sorted(set(range(2, n+1)) - set([p*i for p in range(2, n+1) for i in range(p, n+1)]))And there we have it. The Sieve of Eratosthenes in one line of python. If you'd rather watch the refactoring happening step by step. Here's a video, set to suitable music.

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