Last night on #clojure Lau_of_DK asked for a way to define the sequence of prime numbers.

Having helped Lou Franco in his effort to parallelize primes computation and solved the fourth question of Google Treasure Hunt using Clojure, I thought I knew pretty well how to produce primes in Clojure but I stumbled accross some Haskell code that was far smarter. Here it is, now ported to Clojure:

(def primes (lazy-cons 2 ((fn this[n]
  (let [potential-divisors (take-while #(<= (* % %) n) primes)]
    (if (some #(zero? (rem n %)) potential-divisors) 
      (recur (inc n))
      (lazy-cons n (this (inc n)))))) 3)))

It’s interesting to note that the seq is seeded with 1 and 2 because Clojure’s lazy seqs have a off by one evaluation (when one asks for the nth value, the nth+1 is computed — to know if the end of the seq is reached). No, no, no! I was plain wrong: if I need to seed with [1 2] 2 it’s because of the take-while whose predicate must return at least one false.

Update: In comments, Cale Gibbard points out that my definition of prime numbers is loose: 1 isn’t a prime. I fixed the code.

If I judge by traffic logs for this blog, many newcomers want to compute prime numbers in Clojure.

A recent thread on the mailing list prompted me to test an idea I had for a while: to use a map to implement the Sieve of Eratosthenes.

The first implementation was this one:

(defn primes [max]
  (let [enqueue (fn [sieve n factor]
                  (let [m (+ n factor)]
                    (assoc sieve m
                      (conj (sieve m) factor))))
        next-sieve (fn [sieve candidate]
                     (if-let [factors (sieve candidate)]
                       (reduce #(enqueue %1 candidate %2)
                         (dissoc sieve candidate)
                         factors)
                       (enqueue sieve candidate candidate)))]
    (apply concat (vals (reduce next-sieve {} (range 2 max))))))

where the sieve is a map from the next non-prime numbers to their factors. It’s so naive that even numbers are tested for primality but it doesn’t perform that bad: on my box, it takes 3s to compute all primes below 1,000,000 (it’s roughly as fast as clojure.contrib.lazy-seqs/primes when the seq isn’t yet memoized).

I wasn’t happy with the way I handled the case where a non-prime was already checked off (ie was a key of the map): I was conjing onto the list of prime factors for this number. If instead I tried to check off n+p (where n is the already known non-prime and p the current prime) or n+2p or n+3p… until I found a yet unknown non-prime I wouldn’t need to maintain a list of factors, nor to conj or reduce over it. And I was hoping that less allocations would yield better perfs.

Here is the second iteration:

(defn primes2 [max]
  (let [enqueue (fn [sieve n factor]
                  (let [m (+ n factor)]
                    (if (sieve m)
                      (recur sieve m factor)
                      (assoc sieve m factor))))
        next-sieve (fn [sieve candidate]
                     (if-let [factor (sieve candidate)]
                       (-> sieve
                         (dissoc candidate)
                         (enqueue candidate factor))
                       (enqueue sieve candidate candidate)))]
    (vals (reduce next-sieve {} (range 2 max)))))

and it computes all the primes below 1,000,000 in 1.8s instead of 3s but it still tests even numbers for primality.

primes3 is primes2 modified to only test odd numbers:

(defn primes3 [max]
  (let [enqueue (fn [sieve n factor]
                  (let [m (+ n (+ factor factor))]
                    (if (sieve m)
                      (recur sieve m factor)
                      (assoc sieve m factor))))
        next-sieve (fn [sieve candidate]
                     (if-let [factor (sieve candidate)]
                       (-> sieve
                         (dissoc candidate)
                         (enqueue candidate factor))
                       (enqueue sieve candidate candidate)))]
    (cons 2 (vals (reduce next-sieve {} (range 3 max 2))))))

and it computes the same list of primes in 1.5s.

Out of curiosity, I wrote a lazy version of primes3:

(defn lazy-primes3 []
  (letfn [(enqueue [sieve n step]
            (let [m (+ n step)]
              (if (sieve m)
                (recur sieve m step)
                (assoc sieve m step))))
          (next-sieve [sieve candidate]
            (if-let [step (sieve candidate)]
              (-> sieve
                (dissoc candidate)
                (enqueue candidate step))
              (enqueue sieve candidate (+ candidate candidate))))
          (next-primes [sieve candidate]
            (if (sieve candidate)
              (recur (next-sieve sieve candidate) (+ candidate 2))
              (cons candidate 
                (lazy-seq (next-primes (next-sieve sieve candidate) 
                            (+ candidate 2))))))]
    (cons 2 (lazy-seq (next-primes {} 3)))))

and, surprisingly (better locality?), it computes the primes below 1,000,000 in 1s.

A friend of mine asked me how I solved the fourth question of Google Treasure Hunt 2008 using Clojure. I didn’t keep the original code around, so below is how I could have done it.

First, define a primes seq.

Second, define a function which returns the sequence of sums of N consecutive primes:

(defn sum-primes [n] (map #(apply + %) (partition n 1 primes)))

Third, define a function which, taking a list of increasing sequences, returns the first common value.

(defn find= [seqs]
  (if (apply == (map first seqs))
    (ffirst seqs)
    (let [[s1 & etc] (sort #(- (first %1) (first %2)) seqs)]
      (recur (cons (rest s1) etc)))))

Last, use them! Here is a sample question:

Find the smallest number that can be expressed as
the sum of 3 consecutive prime numbers,
the sum of 5 consecutive prime numbers,
the sum of 11 consecutive prime numbers,
the sum of 1493 consecutive prime numbers,
and is itself a prime number.

And here is how to compute the answer:

(find= (cons primes (map sum-primes [3, 5, 11, 1493])))

returns 9174901 in twenty seconds or so.

(Right now this code may throw a StackOverflow exception, please use one of those definition of partition.)