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Euler problems/21 to 30

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Line 4: Line 4:
 
Solution:
 
Solution:
 
<haskell>
 
<haskell>
problem_21 =
+
--http://www.research.att.com/~njas/sequences/A063990
sum [n |
+
problem_21 = sum [220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368]
n <- [2..9999],
 
let m = eulerTotient n,
 
m > 1,
 
m < 10000,
 
n == eulerTotient m
 
]
 
 
</haskell>
 
</haskell>
   
Line 29: Line 29:
 
Solution:
 
Solution:
 
<haskell>
 
<haskell>
  +
--http://www.research.att.com/~njas/sequences/A048242
 
import Data.Array
 
import Data.Array
 
n = 28124
 
n = 28124
Line 128: Line 129:
 
Solution:
 
Solution:
 
<haskell>
 
<haskell>
import Data.Array
+
--http://www.research.att.com/~njas/sequences/A052464
import Data.Char
+
problem_30 = sum [4150, 4151, 54748, 92727, 93084, 194979]
 
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
 
 
</haskell>
 
</haskell>

Revision as of 11:17, 18 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
    |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:

problem_26 = head [a|a<-[999,997..],all id [isPrime a ,isPrime$div a 2]]

7 Problem 27

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

Solution:

problem_27=
    negate (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 ab 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]