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::::<math>\textstyle\mathbb{S} = \mathbb{N}_{2} \setminus \bigcup_{p\in \mathbb{S}} \{n p:n \in \mathbb{N}_{p}\}</math>
::::<math>\textstyle\mathbb{S} = \mathbb{N}_{2} \setminus \bigcup_{p\in \mathbb{S}} \{n p:n \in \mathbb{N}_{p}\}</math>
-
 
where
where
-
 
::::<math>\textstyle\mathbb{N}_{k} = \{ n \in \mathbb{N} : n \geq k \}</math> &emsp; . . . or, &ensp;<math>\textstyle\mathbb{N}_{k} = \{k\} \bigcup \mathbb{N}_{k+1}</math> &emsp; :)&emsp;:) .
::::<math>\textstyle\mathbb{N}_{k} = \{ n \in \mathbb{N} : n \geq k \}</math> &emsp; . . . or, &ensp;<math>\textstyle\mathbb{N}_{k} = \{k\} \bigcup \mathbb{N}_{k+1}</math> &emsp; :)&emsp;:) .
Trial division sieve:
Trial division sieve:
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::::<math>\textstyle\mathbb{T} = \{n \in \mathbb{N}_{2}: (\not\exists p \in \mathbb{T}) (p\leq \sqrt{n}\,, p\mid n)\}</math>
+
::::<math>\textstyle\mathbb{T} = \{n \in \mathbb{N}_{2}: (\forall p \in \mathbb{T})(2\leq p\leq \sqrt{n}\, \Rightarrow \neg{(p \mid n)})\}</math>
If you're put off by self-referentiality, just replace <math>\mathbb{S}</math> or <math>\mathbb{T}</math> on the right-hand side of equations with <math>\mathbb{N}_{2}</math>.
If you're put off by self-referentiality, just replace <math>\mathbb{S}</math> or <math>\mathbb{T}</math> on the right-hand side of equations with <math>\mathbb{N}_{2}</math>.

Revision as of 18:42, 5 September 2011

I'm interested in Haskell.

I like this: 

--   inifinte folding idea due to Richard Bird
--   double staged production idea due to Melissa O'Neill
--   tree folding idea Dave Bayer / simplified formulation Will Ness
primes = 2 : g (fix g) 
  where                
    g xs = 3 : gaps 5 (foldi (\(c:cs) -> (c:) . union cs) 
                             [[x*x, x*x+2*x..] | x <- xs])
    gaps k s@(c:t)                                        
       | k < c = k : gaps (k+2) s     -- minus [k,k+2..] (c:t), k<=c
       | True  =     gaps (k+2) t     --   fused to avoid a space leak
 
fix g = xs where xs = g xs            -- global defn to avoid space leak

foldi is on Tree-like folds page. union and more at Prime numbers.

The math formula for Sieve of Eratosthenes,

\textstyle\mathbb{S} = \mathbb{N}_{2} \setminus \bigcup_{p\in \mathbb{S}} \{n p:n \in \mathbb{N}_{p}\}

where

\textstyle\mathbb{N}_{k} = \{ n \in \mathbb{N} : n \geq k \}   . . . or,  \textstyle\mathbb{N}_{k} = \{k\} \bigcup \mathbb{N}_{k+1}   :) :) .

Trial division sieve:

\textstyle\mathbb{T} = \{n \in \mathbb{N}_{2}: (\forall p \in \mathbb{T})(2\leq p\leq \sqrt{n}\, \Rightarrow \neg{(p \mid n)})\}

If you're put off by self-referentiality, just replace \mathbb{S} or \mathbb{T} on the right-hand side of equations with \mathbb{N}_{2}.

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