# Euler problems/21 to 30

## Contents

## Problem 21

Evaluate the sum of all amicable pairs under 10000.

Solution:

```
problem_21 =
sum [n |
n <- [2..9999],
let m = eulerTotient n,
m > 1,
m < 10000,
n == eulerTotient m
]
```

## 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 $ show $ sum $ scores
where
score w i = (i *) $ sum $ map (\c -> ord c - ord 'A' + 1) w
```

## Problem 23

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

Solution:

```
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 = putStrLn $ show $ foldl1 (+) $ filter (not . isSum) [1..n]
```

## 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
```

## Problem 25

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

Solution:

```
import Data.List
fib x
|x==0=0
|x==1=1
|x==2=1
|odd x=(fib (d+1))^2+(fib d)^2
|otherwise=(fib (d+1))^2-(fib (d-1))^2
where
d=div x 2
phi=(1+sqrt 5)/2
dig x=floor( (fromInteger x-1) * log 10 /log phi)
problem_25 =
head[a|a<-[dig num..],(>=limit)$fib a]
where
num=1000
limit=10^(num-1)
```

## Problem 26

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

Solution:

```
problem_26 = head [a|a<-[999,997..],all id [isPrime a ,isPrime$div a 2]]
```

## Problem 27

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

Solution:

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

## 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
```

## Problem 29

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

Solution:

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

## 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.Array
import Data.Char
p = listArray (0,9) $ map (^5) [0..9]
upperLimit = 295277
candidates =
[ n |
n <- [10..upperLimit],
(sum $ digits n) `mod` 10 == last(digits n),
powersum n == n
]
where
digits n = map digitToInt $ show n
powersum n = sum $ map (p!) $ digits n
problem_30 = sum candidates
```