Yan Cui

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#### Problem

The arithmetic sequence, 1487, 4817, 8147, in which each of the terms increases by 3330, is unusual in two ways: (i) each of the three terms are prime, and, (ii) each of the 4-digit numbers are permutations of one another.

There are no arithmetic sequences made up of three 1-, 2-, or 3-digit primes, exhibiting this property, but there is one other 4-digit increasing sequence.

What 12-digit number do you form by concatenating the three terms in this sequence?

#### Solution

open System.Collections let rec distribute e = function | [] -> [[e]] | x::xs' as xs -> (e::xs)::[for xs in distribute e xs' -> x::xs] let rec permute = function | [] -> [[]] | e::xs -> List.collect (distribute e) (permute xs) let rec comb n l = match n, l with | 0, _ -> [[]] | _, [] -> [] | k, (x::xs) -> List.map ((@) [x]) (comb (k-1) xs) @ comb k xs let max = 9999 // define a cache for holding records of which number is a prime let cache = new BitArray(max+1, true) // using prime sieve to fill out the cache [2..max] |> List.iter (fun n -> if cache.[n] then [2..max] |> Seq.takeWhile (fun m -> n * m <= max) |> Seq.iter (fun m -> cache.[n * m] List.filter (fun n -> cache.[n]) // define function to get the 4-digit prime permutations of a number let getPrimePermutations n = let digitsStr = n.ToString().ToCharArray() |> Array.map string Array.toList digitsStr |> permute |> Seq.distinct |> Seq.map (fun chars -> int(chars |> List.reduce (+))) |> Seq.filter (fun x -> x >= 1000 && cache.[x]) |> Seq.sort |> Seq.toList let answer = primeNumbers |> List.map getPrimePermutations |> List.filter (fun l -> l |> List.length >= 3) |> Seq.distinct |> Seq.toList |> List.map (fun l -> comb 3 l |> List.filter (fun l' -> l'.[1] - l'.[0] = l'.[2] - l'.[1])) |> List.filter (fun l -> l |> List.length > 0)

The above solution returns two lists:

*val answer : int list list list = [[[1487; 4817; 8147]]; [[2969; 6299; 9629]]]*

The first of course, correlates to the example given in the brief, the other, is the base for your answer!

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