Euler problems/21 to 30

Problem 21

Evaluate the sum of all amicable numbers (including those with a pair number over the limit) 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..]

Here is another alternative solution that computes the sum-of-divisors for the numbers by iterating over products of their factors (very fast):

import Data.Array

max_ = 100000

gen 100001 = []
gen n = [(i*n,n)|i <- [2 .. max_ `div` n]] ++ (gen (n+1))

arr = accumArray (+) 0 (0,max_) (gen 1)

problem_21_v3 = sum $ filter (\a -> let b = (arr!a) in b /= a && (arr!b) == a) [1 .. (10000 - 1)]

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 . 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 = print . sum . 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

Or, using Data.List.permutations,

import Data.List
problem_24 = (!! 999999) . sort $ permutations ['0'..'9']

Casey Hawthorne

For Project Euler #24 you don't need to generate all the lexicographic permutations by Knuth's method or any other.

You're only looking for the millionth lexicographic permutation of "0123456789"

-- Plan of attack.

-- The "x"s are different numbers
-- 0xxxxxxxxx represents 9! = 362880 permutations/numbers
-- 1xxxxxxxxx represents 9! = 362880 permutations/numbers
-- 2xxxxxxxxx represents 9! = 362880 permutations/numbers


-- 20xxxxxxxx represents 8! = 40320
-- 21xxxxxxxx represents 8! = 40320

-- 23xxxxxxxx represents 8! = 40320
-- 24xxxxxxxx represents 8! = 40320
-- 25xxxxxxxx represents 8! = 40320
-- 26xxxxxxxx represents 8! = 40320
-- 27xxxxxxxx represents 8! = 40320


module Euler where

import Data.List

factorial n = product [1..n]

-- lexOrder "0123456789" 1000000 ""

lexOrder digits left s
    | len == 0              = s ++ digits
    | quot > 0 && rem == 0  = lexOrder (digits\\(show (digits!!(quot-1))))  rem (s ++ [(digits!!(quot-1))])
    | quot == 0 && rem == 0 = lexOrder (digits\\(show (digits!!len)))       rem (s ++ [(digits!!len)])
    | rem == 0              = lexOrder (digits\\(show (digits!!(quot+1))))  rem (s ++ [(digits!!(quot+1))])
    | otherwise             = lexOrder (digits\\(show (digits!!(quot))))    rem (s ++ [(digits!!(quot))])
    where
    len = (length digits) - 1
    (quot,rem) = quotRem left (factorial len)

Problem 25

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

Solution:

fibs = 0:1:(zipWith (+) fibs (tail fibs))
t = 10^999

problem_25 = length w
    where
      w = takeWhile (< t) fibs

Casey Hawthorne

I believe you mean the following:

fibs = 0:1:(zipWith (+) fibs (tail fibs))

last (takeWhile (<10^1000) fibs)

Problem 26

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

Solution:

problem_26 = fst $ maximumBy (comparing snd)
                            [(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 elemIndex r' rs of
                                    Just i  -> i + 1
                                    Nothing -> remainders d (10*r') (r':rs)

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

Alternatively, one can use the fact that the distance between the diagonal numbers increases by 2 in every concentric square. Each square contains four gaps, so the following scanl does the trick:

euler28 n = sum $ scanl (+) 0
            (1:(concatMap (replicate 4) [2,4..(n-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]

We can also solve it in a more naive way, without using Monads, like this:

import List
problem_29 = length $ nub pr29_help
    where pr29_help  = [z | y <- [2..100],
                        z <- lift y]
          lift y = map (\x -> x^y) [2..100]

Simpler:

import List
problem_29 = length $ nub [x^y | x <- [2..100], y <- [2..100]]

Instead of using lists, the Set data structure can be used for a significant speed increase:

import Set
problem_29 = size $ fromList [x^y | x <- [2..100], y <- [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.Char (digitToInt)

limit :: Integer
limit = snd $ head $ dropWhile (\(a,b) -> a > b) $ zip (map (9^5*) [1..]) (map (10^) [1..])

fifth :: Integer -> Integer
fifth = sum . map ((^5) . toInteger . digitToInt) . show

problem_30 :: Integer
problem_30 = sum $ filter (\n -> n == fifth n) [2..limit]