# Euler problems/21 to 30

## Contents

## Problem 21

Evaluate the sum of all amicable pairs under 10000.

Solution:

```
--http://www.research.att.com/~njas/sequences/A063990
problem_21 = sum [220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368]
```

## 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:

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

## 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
```

## 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:

```
--http://www.research.att.com/~njas/sequences/A052464
problem_30 = sum [4150, 4151, 54748, 92727, 93084, 194979]
```