# Euler problems/21 to 30

### From HaskellWiki

(Fixed version 2 to have proper solution. Numbers whose divisor sum equals themselves are not amicable numbers.) |
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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) | (http://www.research.att.com/~njas/sequences/A063990) | ||

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<haskell> | <haskell> | ||

problem_21_v2 = sum [n | n <- [2..9999], let m = d n, | problem_21_v2 = sum [n | n <- [2..9999], let m = d n, | ||

− | m > 1, m < 10000, n == d m] | + | m > 1, m < 10000, n == d m, d m /= d (d m)] |

d n = product [(p * product g - 1) `div` (p - 1) | | d n = product [(p * product g - 1) `div` (p - 1) | | ||

g <- group $ primeFactors n, let p = head g | g <- group $ primeFactors n, let p = head g |

## Revision as of 21:25, 21 October 2008

## Contents |

## 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..]

## 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:

valid ( i, n ) = length ( show n ) == 1000 problem_25 = fst . head . filter valid . zip [ 1 .. ] $ fibs where fibs = 1 : 1 : 2 : zipWith (+) fibs ( tail fibs )

## 6 Problem 26

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

Solution:

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)

## 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:

import Data.Char (ord) limit :: Integer limit = snd $ head $ dropWhile (\(a,b) -> a > b) $ zip (map (9^5*) [1..]) (map (10^) [1..]) fifth :: Integer -> Integer fifth n = foldr (\a b -> (toInteger(ord a) - 48)^5 + b) 0 $ show n problem_30 :: Integer problem_30 = sum $ filter (\n -> n == fifth n) [2..limit]