# Euler problems/21 to 30

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## Revision as of 13:46, 22 January 2008

## Contents |

## 1 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 ]

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

## 3 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]

## 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

## 5 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)

## 6 Problem 26

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

Solution:

next n d = (n `mod` d):next (10*n`mod`d) d idigs n = tail $ take (1+n) $ next 1 n pos x = map fst . filter ((==x) . snd) . zip [1..] periods n = let d = idigs n in pos (head d) (tail d) problem_26 = snd$maximum [(m,a)| a<-[800..1000] , let k=periods a, not$null k, let m=head k ]

## 7 Problem 27

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

Solution:

eulerCoefficients n = [((len, a*b), (a, b)) | b <- takeWhile (<n) primes, a <- [-b+1..n-1], let len = length $ takeWhile (isPrime . (\x -> x^2 + a*x + b)) [0..], if b == 2 then even a else odd a, len > 39] problem_27 = snd . fst . maximum . eulerCoefficients $ 1000

## 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

## 9 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]

## 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.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