Euler problems/21 to 30
m (Added a simpler solution to problem 29)
Revision as of 17:33, 8 November 2010
Evaluate the sum of all amicable pairs under 10000.
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..]
What is the total of all the name scores in the file of first names?
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
Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.
--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]
What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9?
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
What is the first term in the Fibonacci sequence to contain 1000 digits?
fibs = 0:1:(zipWith (+) fibs (tail fibs)) t = 10^999 problem_25 = length w where w = takeWhile (< t) fibs
Find the value of d < 1000 for which 1/d contains the longest recurring cycle.
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)
Find a quadratic formula that produces the maximum number of primes for consecutive values of n.
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
What is the sum of both diagonals in a 1001 by 1001 spiral?
problem_28 = sum (map (\n -> 4*(n-2)^2+10*(n-1)) [3,5..1001]) + 1
How many distinct terms are in the sequence generated by ab for 2 ≤ a ≤ 100 and 2 ≤ b ≤ 100?
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]
import List problem_29 = length $ nub [x^y | x <- [2..100], y <- [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.
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]