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[[Category:Programming exercise spoilers]]
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== [http://projecteuler.net/index.php?section=problems&id=21 Problem 21] ==
== [http://projecteuler.net/index.php?section=view&id=21 Problem 21] ==
 
 
Evaluate the sum of all amicable pairs under 10000.
 
Evaluate the sum of all amicable pairs under 10000.
   
Solution:
+
Solution:
  +
(http://www.research.att.com/~njas/sequences/A063990)
  +
 
This is a little slow because of the naive method used to compute the divisors.
 
This is a little slow because of the naive method used to compute the divisors.
 
<haskell>
 
<haskell>
Line 12: Line 12:
 
</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=22 Problem 22] ==
+
Here is an alternative using a faster way of computing the sum of divisors.
  +
<haskell>
  +
problem_21_v2 = sum [n | n <- [2..9999], let m = d n,
  +
m > 1, m < 10000, n == d m, d m /= d (d m)]
  +
d n = product [(p * product g - 1) `div` (p - 1) |
  +
g <- group $ primeFactors n, let p = head g
  +
] - n
  +
primeFactors = pf primes
  +
where
  +
pf ps@(p:ps') n
  +
| p * p > n = [n]
  +
| r == 0 = p : pf ps q
  +
| otherwise = pf ps' n
  +
where (q, r) = n `divMod` p
  +
primes = 2 : filter (null . tail . primeFactors) [3,5..]
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=22 Problem 22] ==
 
What is the total of all the name scores in the file of first names?
 
What is the total of all the name scores in the file of first names?
   
 
Solution:
 
Solution:
 
<haskell>
 
<haskell>
-- apply to a list of names
+
import Data.List
problem_22 :: [String] -> Int
+
import Data.Char
problem_22 = sum . zipWith (*) [ 1 .. ] . map score
+
problem_22 =
where score = sum . map ( subtract 64 . ord )
+
do input <- readFile "names.txt"
  +
let names = sort $ read$"["++ input++"]"
  +
let scores = zipWith score names [1..]
  +
print . sum $ scores
  +
where score w i = (i *) . sum . map (\c -> ord c - ord 'A' + 1) $ w
 
</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=23 Problem 23] ==
+
== [http://projecteuler.net/index.php?section=problems&id=23 Problem 23] ==
 
Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.
 
Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.
   
 
Solution:
 
Solution:
 
<haskell>
 
<haskell>
import Data.Set hiding (filter, map)
+
--http://www.research.att.com/~njas/sequences/A048242
import Data.List (scanl, group)
+
import Data.Array
  +
n = 28124
  +
abundant n = eulerTotient n - n > n
  +
abunds_array = listArray (1,n) $ map abundant [1..n]
  +
abunds = filter (abunds_array !) [1..n]
   
problem_23 :: Integer
+
rests x = map (x-) $ takeWhile (<= x `div` 2) abunds
problem_23 = sum [1..28123] - (fold (+) 0 $ abundant_sums $ abundant 28123)
+
isSum = any (abunds_array !) . rests
   
abundant_sums :: [Integer] -> Set Integer
+
problem_23 = print . sum . filter (not . isSum) $ [1..n]
abundant_sums [] = empty
 
abundant_sums l@(x:xs) = union (fromList [x + a | a <- takeWhile (\y -> y <= 28123 - x) l]) (abundant_sums xs)
 
 
abundant :: Integer -> [Integer]
 
abundant n = [a | a <- [1..n], (sum $ factors a) - a > a]
 
 
primes :: [Integer]
 
primes = 2 : filter ((==1) . length . primeFactors) [3,5..]
 
 
primeFactors :: Integer -> [Integer]
 
primeFactors n = factor n primes
 
where
 
factor _ [] = []
 
factor m (p:ps) | p*p > m = [m]
 
| m `mod` p == 0 = p : factor (m `div` p) (p:ps)
 
| otherwise = factor m ps
 
 
factors :: Integer -> [Integer]
 
factors = perms . map (tail . scanl (*) 1) . group . primeFactors
 
where
 
perms :: (Integral a) => [[a]] -> [a]
 
perms [] = [1]
 
perms (x:xs) = perms xs ++ concatMap (\z -> map (*z) $ perms xs) x
 
 
</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=24 Problem 24] ==
+
== [http://projecteuler.net/index.php?section=problems&id=24 Problem 24] ==
 
What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9?
 
What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9?
   
 
Solution:
 
Solution:
 
<haskell>
 
<haskell>
perms [] = [[]]
+
import Data.List
perms xs = do
+
x <- xs
+
fac 0 = 1
map ( x: ) ( perms . delete x $ xs )
+
fac n = n * fac (n - 1)
+
perms [] _= []
problem_24 = ( perms "0123456789" ) !! 999999
+
perms xs n= x : perms (delete x xs) (mod n m)
  +
where m = fac $ length xs - 1
  +
y = div n m
  +
x = xs!!y
  +
  +
problem_24 = perms "0123456789" 999999
 
</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=25 Problem 25] ==
+
Or, using Data.List.permutations,
What is the first term in the Fibonacci sequence to contain 1000 digits?
 
 
Solution:
 
 
<haskell>
 
<haskell>
valid ( i, n ) = length ( show n ) == 1000
+
import Data.List
+
problem_24 = (!! 999999) . sort $ permutations ['0'..'9']
problem_25 = fst . head . filter valid . zip [ 1 .. ] $ fibs
 
where fibs = 1 : 1 : 2 : zipWith (+) fibs ( tail fibs )
 
 
</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=26 Problem 26] ==
+
Casey Hawthorne
Find the value of d < 1000 for which 1/d contains the longest recurring cycle.
+
  +
For Project Euler #24 you don't need to generate all the lexicographic permutations by Knuth's method or any other.
  +
  +
You're only looking for the millionth lexicographic permutation of "0123456789"
   
Solution:
 
 
<haskell>
 
<haskell>
problem_26 = fst $ maximumBy (\a b -> snd a `compare` snd b)
 
[(n,recurringCycle n) | n <- [1..999]]
 
where recurringCycle d = remainders d 10 []
 
remainders d 0 rs = 0
 
remainders d r rs = let r' = r `mod` d
 
in case findIndex (== r') rs of
 
Just i -> i + 1
 
Nothing -> remainders d (10*r') (r':rs)
 
</haskell>
 
   
== [http://projecteuler.net/index.php?section=view&id=27 Problem 27] ==
+
-- Plan of attack.
Find a quadratic formula that produces the maximum number of primes for consecutive values of n.
 
   
Solution:
+
-- The "x"s are different numbers
  +
-- 0xxxxxxxxx represents 9! = 362880 permutations/numbers
  +
-- 1xxxxxxxxx represents 9! = 362880 permutations/numbers
  +
-- 2xxxxxxxxx represents 9! = 362880 permutations/numbers
   
The following is written in [http://haskell.org/haskellwiki/Literate_programming#Haskell_and_literate_programming literate Haskell]:
 
<haskell>
 
> import Data.List
 
   
To be sure we get the maximum type checking of the compiler,
+
-- 20xxxxxxxx represents 8! = 40320
we switch off the default type
+
-- 21xxxxxxxx represents 8! = 40320
   
> default ()
+
-- 23xxxxxxxx represents 8! = 40320
  +
-- 24xxxxxxxx represents 8! = 40320
  +
-- 25xxxxxxxx represents 8! = 40320
  +
-- 26xxxxxxxx represents 8! = 40320
  +
-- 27xxxxxxxx represents 8! = 40320
   
Generate a list of primes.
 
It works by filtering out numbers that are
 
divisable by a previously found prime
 
   
> primes :: [Int]
+
module Euler where
> primes = sieve (2 : [3, 5..])
 
> where
 
> sieve (p:xs) = p : sieve (filter (\x -> x `mod` p > 0) xs)
 
   
> isPrime :: Int -> Bool
+
import Data.List
> isPrime x = x `elem` (takeWhile (<= x) primes)
 
   
  +
factorial n = product [1..n]
   
The lists of values we are going to try for a and b;
+
-- lexOrder "0123456789" 1000000 ""
b must be a prime, as n² + an + b is equal to b when n = 0
 
   
> testRangeA :: [Int]
+
lexOrder digits left s
> testRangeA = [-1000 .. 1000]
+
| len == 0 = s ++ digits
  +
| quot > 0 && rem == 0 = lexOrder (digits\\(show (digits!!(quot-1)))) rem (s ++ [(digits!!(quot-1))])
  +
| quot == 0 && rem == 0 = lexOrder (digits\\(show (digits!!len))) rem (s ++ [(digits!!len)])
  +
| rem == 0 = lexOrder (digits\\(show (digits!!(quot+1)))) rem (s ++ [(digits!!(quot+1))])
  +
| otherwise = lexOrder (digits\\(show (digits!!(quot)))) rem (s ++ [(digits!!(quot))])
  +
where
  +
len = (length digits) - 1
  +
(quot,rem) = quotRem left (factorial len)
   
> testRangeB :: [Int]
+
</haskell>
> testRangeB = takeWhile (< 1000) primes
 
   
  +
== [http://projecteuler.net/index.php?section=problems&id=25 Problem 25] ==
  +
What is the first term in the Fibonacci sequence to contain 1000 digits?
   
The search
+
Solution:
  +
<haskell>
  +
fibs = 0:1:(zipWith (+) fibs (tail fibs))
  +
t = 10^999
   
> bestCoefficients :: (Int, Int, Int)
+
problem_25 = length w
> bestCoefficients =
+
where
> maximumBy (\(x, _, _) (y, _, _) -> compare x y) $
+
w = takeWhile (< t) fibs
> [f a b | a <- testRangeA, b <- testRangeB]
+
</haskell>
> where
 
   
Generate a list of results of the quadratic formula
 
(only the contiguous primes)
 
wrap the result in a triple, together with a and b
 
   
> f :: Int -> Int -> (Int, Int, Int)
+
Casey Hawthorne
> f a b = ( length $ contiguousPrimes a b
 
> , a
 
> , b
 
> )
 
   
> contiguousPrimes :: Int -> Int -> [Int]
+
I believe you mean the following:
> contiguousPrimes a b = takeWhile isPrime (map (quadratic a b) [0..])
 
   
  +
<haskell>
   
The quadratic formula
+
fibs = 0:1:(zipWith (+) fibs (tail fibs))
   
> quadratic :: Int -> Int -> Int -> Int
+
last (takeWhile (<10^1000) fibs)
> quadratic a b n = n * n + a * n + b
+
</haskell>
   
  +
== [http://projecteuler.net/index.php?section=problems&id=26 Problem 26] ==
  +
Find the value of d < 1000 for which 1/d contains the longest recurring cycle.
   
> problem_27 =
+
Solution:
> do
+
<haskell>
> let (l, a, b) = bestCoefficients
+
problem_26 = fst $ maximumBy (comparing snd)
>
+
[(n,recurringCycle n) | n <- [1..999]]
> putStrLn $ ""
+
where recurringCycle d = remainders d 10 []
> putStrLn $ "Problem Euler 27"
+
remainders d 0 rs = 0
> putStrLn $ ""
+
remainders d r rs = let r' = r `mod` d
> putStrLn $ "The best quadratic formula found is:"
+
in case elemIndex r' rs of
> putStrLn $ " n * n + " ++ show a ++ " * n + " ++ show b
+
Just i -> i + 1
> putStrLn $ ""
+
Nothing -> remainders d (10*r') (r':rs)
> putStrLn $ "The number of primes is: " ++ (show l)
+
</haskell>
> putStrLn $ ""
 
> putStrLn $ "The primes are:"
 
> print $ take l $ contiguousPrimes a b
 
> putStrLn $ ""
 
   
  +
== [http://projecteuler.net/index.php?section=problems&id=27 Problem 27] ==
  +
Find a quadratic formula that produces the maximum number of primes for consecutive values of n.
   
  +
Solution:
  +
<haskell>
  +
problem_27 = -(2*a-1)*(a^2-a+41)
  +
where n = 1000
  +
m = head $ filter (\x->x^2-x+41>n) [1..]
  +
a = m-1
 
</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=28 Problem 28] ==
+
== [http://projecteuler.net/index.php?section=problems&id=28 Problem 28] ==
 
What is the sum of both diagonals in a 1001 by 1001 spiral?
 
What is the sum of both diagonals in a 1001 by 1001 spiral?
   
 
Solution:
 
Solution:
 
<haskell>
 
<haskell>
corners :: Int -> (Int, Int, Int, Int)
+
problem_28 = sum (map (\n -> 4*(n-2)^2+10*(n-1)) [3,5..1001]) + 1
corners i = (n*n, 1+(n*(2*m)), 2+(n*(2*m-1)), 3+(n*(2*m-2)))
+
</haskell>
where m = (i-1) `div` 2
 
n = 2*m+1
 
   
sumcorners :: Int -> Int
+
Alternatively, one can use the fact that the distance between the diagonal numbers increases by 2 in every concentric square. Each square contains four gaps, so the following <hask>scanl</hask> does the trick:
sumcorners i = a+b+c+d where (a, b, c, d) = corners i
 
   
sumdiags :: Int -> Int
+
<haskell>
sumdiags i | even i = error "not a spiral"
+
euler28 n = sum $ scanl (+) 0
| i == 3 = s + 1
+
(1:(concatMap (replicate 4) [2,4..(n-1)]))
| otherwise = s + sumdiags (i-2)
 
where s = sumcorners i
 
 
problem_28 = sumdiags 1001
 
 
</haskell>
 
</haskell>
   
You can note that from 1 to 3 there's (+2), and such too for 5, 7 and 9, it then goes up to (+4) 4 times, and so on, adding 2 to the number to add for each level of the spiral. You can so avoid all need for multiplications and just do additions with the following code :
+
== [http://projecteuler.net/index.php?section=problems&id=29 Problem 29] ==
<haskell>problem_28 = sum . scanr ($) 1 . concatMap (replicate 4 . (+)) $ [1000,998..2]</haskell>
 
 
== [http://projecteuler.net/index.php?section=view&id=29 Problem 29] ==
 
 
How many distinct terms are in the sequence generated by a<sup>b</sup> for 2 ≤ a ≤ 100 and 2 ≤ b ≤ 100?
 
How many distinct terms are in the sequence generated by a<sup>b</sup> for 2 ≤ a ≤ 100 and 2 ≤ b ≤ 100?
   
 
Solution:
 
Solution:
 
<haskell>
 
<haskell>
problem_29 = length . group . sort $ [a^b | a <- [2..100], b <- [2..100]]
+
import Control.Monad
  +
problem_29 = length . group . sort $ liftM2 (^) [2..100] [2..100]
  +
</haskell>
  +
  +
We can also solve it in a more naive way, without using Monads, like this:
  +
<haskell>
  +
import List
  +
problem_29 = length $ nub pr29_help
  +
where pr29_help = [z | y <- [2..100],
  +
z <- lift y]
  +
lift y = map (\x -> x^y) [2..100]
  +
</haskell>
  +
  +
Simpler:
  +
  +
<haskell>
  +
import List
  +
problem_29 = length $ nub [x^y | x <- [2..100], y <- [2..100]]
  +
</haskell>
  +
  +
Instead of using lists, the Set data structure can be used for a significant speed increase:
  +
  +
<haskell>
  +
import Set
  +
problem_29 = size $ fromList [x^y | x <- [2..100], y <- [2..100]]
 
</haskell>
 
</haskell>
   
== [http://projecteuler.net/index.php?section=view&id=30 Problem 30] ==
+
== [http://projecteuler.net/index.php?section=problems&id=30 Problem 30] ==
 
Find the sum of all the numbers that can be written as the sum of fifth powers of their digits.
 
Find the sum of all the numbers that can be written as the sum of fifth powers of their digits.
   
 
Solution:
 
Solution:
 
<haskell>
 
<haskell>
import Data.Char (ord)
+
import Data.Char (digitToInt)
   
 
limit :: Integer
 
limit :: Integer
Line 172: Line 167:
   
 
fifth :: Integer -> Integer
 
fifth :: Integer -> Integer
fifth n = foldr (\a b -> (toInteger(ord a) - 48)^5 + b) 0 $ show n
+
fifth = sum . map ((^5) . toInteger . digitToInt) . show
   
 
problem_30 :: Integer
 
problem_30 :: Integer
 
problem_30 = sum $ filter (\n -> n == fifth n) [2..limit]
 
problem_30 = sum $ filter (\n -> n == fifth n) [2..limit]
 
</haskell>
 
</haskell>
 
 
[[Category:Tutorials]]
 
[[Category:Code]]
 

Latest revision as of 03:52, 14 November 2011

Contents

[edit] 1 Problem 21

Evaluate the sum of all amicable pairs under 10000.

Solution: (http://www.research.att.com/~njas/sequences/A063990)

This is a little slow because of the naive method used to compute the divisors.

problem_21 = sum [m+n | m <- [2..9999], let n = divisorsSum ! m, amicable m n]
    where amicable m n = m < n && n < 10000 && divisorsSum ! n == m
          divisorsSum = array (1,9999)
                        [(i, sum (divisors i)) | i <- [1..9999]]
          divisors n = [j | j <- [1..n `div` 2], n `mod` j == 0]

Here is an alternative using a faster way of computing the sum of divisors.

problem_21_v2 = sum [n | n <- [2..9999], let m = d n,
                         m > 1, m < 10000, n == d m, d m /= d  (d m)]
d n = product [(p * product g - 1) `div` (p - 1) |
                 g <- group $ primeFactors n, let p = head g
              ] - n
primeFactors = pf primes
  where
    pf ps@(p:ps') n
     | p * p > n = [n]
     | r == 0    = p : pf ps q
     | otherwise = pf ps' n
     where (q, r) = n `divMod` p
primes = 2 : filter (null . tail . primeFactors) [3,5..]

[edit] 2 Problem 22

What is the total of all the name scores in the file of first names?

Solution:

import Data.List
import Data.Char
problem_22 =
    do input <- readFile "names.txt"
       let names = sort $ read$"["++ input++"]"
       let scores = zipWith score names [1..]
       print . sum $ scores
  where score w i = (i *) . sum . map (\c -> ord c - ord 'A' + 1) $ w

[edit] 3 Problem 23

Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.

Solution:

--http://www.research.att.com/~njas/sequences/A048242
import Data.Array 
n = 28124
abundant n = eulerTotient n - n > n
abunds_array = listArray (1,n) $ map abundant [1..n]
abunds = filter (abunds_array !) [1..n]
 
rests x = map (x-) $ takeWhile (<= x `div` 2) abunds
isSum = any (abunds_array !) . rests
 
problem_23 = print . sum . filter (not . isSum) $ [1..n]

[edit] 4 Problem 24

What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9?

Solution:

import Data.List 
 
fac 0 = 1
fac n = n * fac (n - 1)
perms [] _= []
perms xs n= x : perms (delete x xs) (mod n m)
  where m = fac $ length xs - 1
        y = div n m
        x = xs!!y
 
problem_24 = perms "0123456789" 999999

Or, using Data.List.permutations,

import Data.List
problem_24 = (!! 999999) . sort $ permutations ['0'..'9']

Casey Hawthorne

For Project Euler #24 you don't need to generate all the lexicographic permutations by Knuth's method or any other.

You're only looking for the millionth lexicographic permutation of "0123456789"

-- Plan of attack.
 
-- The "x"s are different numbers
-- 0xxxxxxxxx represents 9! = 362880 permutations/numbers
-- 1xxxxxxxxx represents 9! = 362880 permutations/numbers
-- 2xxxxxxxxx represents 9! = 362880 permutations/numbers
 
 
-- 20xxxxxxxx represents 8! = 40320
-- 21xxxxxxxx represents 8! = 40320
 
-- 23xxxxxxxx represents 8! = 40320
-- 24xxxxxxxx represents 8! = 40320
-- 25xxxxxxxx represents 8! = 40320
-- 26xxxxxxxx represents 8! = 40320
-- 27xxxxxxxx represents 8! = 40320
 
 
module Euler where
 
import Data.List
 
factorial n = product [1..n]
 
-- lexOrder "0123456789" 1000000 ""
 
lexOrder digits left s
    | len == 0              = s ++ digits
    | quot > 0 && rem == 0  = lexOrder (digits\\(show (digits!!(quot-1))))  rem (s ++ [(digits!!(quot-1))])
    | quot == 0 && rem == 0 = lexOrder (digits\\(show (digits!!len)))       rem (s ++ [(digits!!len)])
    | rem == 0              = lexOrder (digits\\(show (digits!!(quot+1))))  rem (s ++ [(digits!!(quot+1))])
    | otherwise             = lexOrder (digits\\(show (digits!!(quot))))    rem (s ++ [(digits!!(quot))])
    where
    len = (length digits) - 1
    (quot,rem) = quotRem left (factorial len)

[edit] 5 Problem 25

What is the first term in the Fibonacci sequence to contain 1000 digits?

Solution:

fibs = 0:1:(zipWith (+) fibs (tail fibs))
t = 10^999
 
problem_25 = length w
    where
      w = takeWhile (< t) fibs


Casey Hawthorne

I believe you mean the following:

fibs = 0:1:(zipWith (+) fibs (tail fibs))
 
last (takeWhile (<10^1000) fibs)

[edit] 6 Problem 26

Find the value of d < 1000 for which 1/d contains the longest recurring cycle.

Solution:

problem_26 = fst $ maximumBy (comparing snd)
                            [(n,recurringCycle n) | n <- [1..999]]
    where  recurringCycle d = remainders d 10 []
           remainders d 0 rs = 0
           remainders d r rs = let r' = r `mod` d
                               in case elemIndex r' rs of
                                    Just i  -> i + 1
                                    Nothing -> remainders d (10*r') (r':rs)

[edit] 7 Problem 27

Find a quadratic formula that produces the maximum number of primes for consecutive values of n.

Solution:

problem_27 = -(2*a-1)*(a^2-a+41)
  where n = 1000
        m = head $ filter (\x->x^2-x+41>n) [1..]
        a = m-1

[edit] 8 Problem 28

What is the sum of both diagonals in a 1001 by 1001 spiral?

Solution:

problem_28 = sum (map (\n -> 4*(n-2)^2+10*(n-1)) [3,5..1001]) + 1
Alternatively, one can use the fact that the distance between the diagonal numbers increases by 2 in every concentric square. Each square contains four gaps, so the following
scanl
does the trick:
euler28 n = sum $ scanl (+) 0
            (1:(concatMap (replicate 4) [2,4..(n-1)]))

[edit] 9 Problem 29

How many distinct terms are in the sequence generated by ab for 2 ≤ a ≤ 100 and 2 ≤ b ≤ 100?

Solution:

import Control.Monad
problem_29 = length . group . sort $ liftM2 (^) [2..100] [2..100]

We can also solve it in a more naive way, without using Monads, like this:

import List
problem_29 = length $ nub pr29_help
    where pr29_help  = [z | y <- [2..100],
                        z <- lift y]
          lift y = map (\x -> x^y) [2..100]

Simpler:

import List
problem_29 = length $ nub [x^y | x <- [2..100], y <- [2..100]]

Instead of using lists, the Set data structure can be used for a significant speed increase:

import Set
problem_29 = size $ fromList [x^y | x <- [2..100], y <- [2..100]]

[edit] 10 Problem 30

Find the sum of all the numbers that can be written as the sum of fifth powers of their digits.

Solution:

import Data.Char (digitToInt)
 
limit :: Integer
limit = snd $ head $ dropWhile (\(a,b) -> a > b) $ zip (map (9^5*) [1..]) (map (10^) [1..])
 
fifth :: Integer -> Integer
fifth = sum . map ((^5) . toInteger . digitToInt) . show
 
problem_30 :: Integer
problem_30 = sum $ filter (\n -> n == fifth n) [2..limit]