# Euler problems/21 to 30

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## Revision as of 19:25, 19 February 2008

## Contents |

## 1 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]

## 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:

--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]

## 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 | 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)

## 6 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]]

## 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

## 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:

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