Números que no son cuadrados

PorJosé A. Alonso 15-febrero-201822-febrero-2018

Definir las funciones

   noCuadrados :: [Integer]
   graficaNoCuadrados :: Integer -> IO ()

1 2	noCuadrados :: [Integer] graficaNoCuadrados :: Integer -> IO ()

tales que

noCuadrados es la lista de los números naturales que no son cuadrados. Por ejemplo,

     λ> take 25 noCuadrados
     [2,3,5,6,7,8,10,11,12,13,14,15,17,18,19,20,21,22,23,24,26,27,28,29,30]

1 2	λ> take 25 noCuadrados [2,3,5,6,7,8,10,11,12,13,14,15,17,18,19,20,21,22,23,24,26,27,28,29,30]

(graficaNoCuadrados n) dibuja las diferencias entre los n primeros elementos de noCuadrados y sus posiciones. Por ejemplo, (graficaNoCuadrados 300) dibuja

(graficaNoCuadrados 3000) dibuja

(graficaNoCuadrados 30000) dibuja

Comprobar con QuickCheck que el término de noCuadrados en la posición n-1 es (n + floor(1/2 + sqrt(n))).

Soluciones

import Data.List (genericIndex)
import Graphics.Gnuplot.Simple
import Test.QuickCheck

-- 1ª definición
-- =============

noCuadrados :: [Integer]
noCuadrados = aux [0..] cuadrados
  where aux xs (y:ys) = as ++ aux bs ys
          where (as,_:bs) = span (<y) xs

cuadrados :: [Integer]
cuadrados = [x^2 | x <- [0..]]

-- 2ª definición
-- =============

noCuadrados2 :: [Integer]
noCuadrados2 = aux 2 [1..]
  where aux n (_:xs) = ys ++ aux (n+2) zs
          where (ys,zs) = splitAt n xs

-- Definición de graficaNoCuadrados
-- ================================

graficaNoCuadrados :: Integer -> IO ()
graficaNoCuadrados n =
  plotList [ Key Nothing
           , PNG ("Numeros_que_no_son_cuadrados_" ++ show n ++ ".png")
           ]
           (zipWith (-) noCuadrados [0..n-1])

-- La propiedad es
prop_noCuadrados :: Positive Integer -> Bool
prop_noCuadrados (Positive n) =
  noCuadrados `genericIndex` (n-1) ==
  n + floor (1/2 + sqrt (fromIntegral n))

-- La comprobación es
--    λ> quickCheck prop_noCuadrados
--    +++ OK, passed 100 tests.

import Data.List (genericIndex)

import Graphics.Gnuplot.Simple

import Test.QuickCheck

-- 1ª definición

-- =============

noCuadrados :: [Integer]

noCuadrados = aux [0..] cuadrados

where aux xs (y:ys) = as ++ aux bs ys

where (as,_:bs) = span (<y) xs

cuadrados :: [Integer]

cuadrados = [x^2 | x <- [0..]]

-- 2ª definición

-- =============

noCuadrados2 :: [Integer]

noCuadrados2 = aux 2 [1..]

where aux n (_:xs) = ys ++ aux (n+2) zs

where (ys,zs) = splitAt n xs

-- Definición de graficaNoCuadrados

-- ================================

graficaNoCuadrados :: Integer -> IO ()

graficaNoCuadrados n =

plotList [ Key Nothing

, PNG ("Numeros_que_no_son_cuadrados_" ++ show n ++ ".png")

]

(zipWith (-) noCuadrados [0..n-1])

-- La propiedad es

prop_noCuadrados :: Positive Integer -> Bool

prop_noCuadrados (Positive n) =

noCuadrados `genericIndex` (n-1) ==

n + floor (1/2 + sqrt (fromIntegral n))

-- La comprobación es

-- λ> quickCheck prop_noCuadrados

-- +++ OK, passed 100 tests.

7 Comentarios

import Data.List               (genericTake, genericIndex)
import Test.QuickCheck
import Graphics.Gnuplot.Simple (plotList, Attribute (Key))

noCuadrados :: [Integer]
noCuadrados = aux 2 [1..]
  where aux n (x:xs) = ys ++ aux (n+2) zs
          where (ys,zs) = splitAt n xs

graficaNoCuadrados :: Integer -> IO ()
graficaNoCuadrados n =
  plotList
  [Key Nothing]
  (zipWith (-) (genericTake n noCuadrados) [0..])

-- Propiedad
-- =========

-- La propiedad es
prop_noCuadrados :: (Positive Integer) -> Bool
prop_noCuadrados (Positive n) =
  genericIndex noCuadrados (n-1)
  == n + floor (1/2 + sqrt (fromIntegral n))

-- La comprobación es
--    λ> quickCheck prop_noCuadrados
--    +++ OK, passed 100 tests.

import Data.List (genericTake, genericIndex)

import Test.QuickCheck

import Graphics.Gnuplot.Simple (plotList, Attribute (Key))

noCuadrados :: [Integer]

noCuadrados = aux 2 [1..]

where aux n (x:xs) = ys ++ aux (n+2) zs

where (ys,zs) = splitAt n xs

graficaNoCuadrados :: Integer -> IO ()

graficaNoCuadrados n =

plotList

[Key Nothing]

(zipWith (-) (genericTake n noCuadrados) [0..])

-- Propiedad

-- =========

-- La propiedad es

prop_noCuadrados :: (Positive Integer) -> Bool

prop_noCuadrados (Positive n) =

genericIndex noCuadrados (n-1)

== n + floor (1/2 + sqrt (fromIntegral n))

-- La comprobación es

-- λ> quickCheck prop_noCuadrados

-- +++ OK, passed 100 tests.

Responder

import Graphics.Gnuplot.Simple
import Test.QuickCheck

noCuadrados :: [Integer]
noCuadrados = aux [2..] 4
  where aux (x:xs) z | x == z    = aux xs ((floor(sqrt(fromIntegral z))+1)^2)
                     | otherwise = x : aux xs z
                     
graficaNoCuadrados :: Integer -> IO ()
graficaNoCuadrados n = plotList [] (aux (take (fromIntegral n) noCuadrados))
  where aux :: [Integer] -> [Integer]
        aux (y:ys) = (1+(floor(sqrt (fromIntegral y)))) : (aux ys)
        aux _ = []

prop_noCuadrados :: Integer -> Property
prop_noCuadrados n = n>0 ==> noCuadrados !! (t-1) ==  (n + (floor (1/2 + sqrt(fromIntegral n))))
  where t = fromIntegral n

-- Y la comprobación es:

-- λ> quickCheck prop_noCuadrados
-- +++ OK, passed 100 tests.
-- (0.01 secs, 3,991,304 bytes)

import Graphics.Gnuplot.Simple

import Test.QuickCheck

noCuadrados :: [Integer]

noCuadrados = aux [2..] 4

where aux (x:xs) z | x == z = aux xs ((floor(sqrt(fromIntegral z))+1)^2)

| otherwise = x : aux xs z

graficaNoCuadrados :: Integer -> IO ()

graficaNoCuadrados n = plotList [] (aux (take (fromIntegral n) noCuadrados))

where aux :: [Integer] -> [Integer]

aux (y:ys) = (1+(floor(sqrt (fromIntegral y)))) : (aux ys)

aux _ = []

prop_noCuadrados :: Integer -> Property

prop_noCuadrados n = n>0 ==> noCuadrados !! (t-1) == (n + (floor (1/2 + sqrt(fromIntegral n))))

where t = fromIntegral n

-- Y la comprobación es:

-- λ> quickCheck prop_noCuadrados

-- +++ OK, passed 100 tests.

-- (0.01 secs, 3,991,304 bytes)

Responder

import Graphics.Gnuplot.Simple
import Test.QuickCheck
import Data.List  (genericTake)

noCuadrados :: [Integer]
noCuadrados = filter (noCuadrado) [2..]

noCuadrado :: Integer -> Bool
noCuadrado n |length a == 1 && even (head a) = False
             |otherwise = True
  where a = (map length (group (primeFactors n)))

graficaNoCuadrados :: Integer -> IO ()
graficaNoCuadrados n = plotList [Title "Numeros que no son cuadrados", 
                                 Key Nothing] 
                       (zipWith (-) (genericTake n noCuadrados) [0..])  

prop_noCuadrados :: Integer -> Property
prop_noCuadrados n = n > 0 ==> noCuadrados !! (fromIntegral n-1) ==  
                               (n + (floor (1/2 + sqrt(fromIntegral n))))

import Graphics.Gnuplot.Simple

import Test.QuickCheck

import Data.List (genericTake)

noCuadrados :: [Integer]

noCuadrados = filter (noCuadrado) [2..]

noCuadrado :: Integer -> Bool

noCuadrado n |length a == 1 && even (head a) = False

|otherwise = True

where a = (map length (group (primeFactors n)))

graficaNoCuadrados :: Integer -> IO ()

graficaNoCuadrados n = plotList [Title "Numeros que no son cuadrados",

Key Nothing]

(zipWith (-) (genericTake n noCuadrados) [0..])

prop_noCuadrados :: Integer -> Property

prop_noCuadrados n = n > 0 ==> noCuadrados !! (fromIntegral n-1) ==

(n + (floor (1/2 + sqrt(fromIntegral n))))

Responder

antgongar dice:

20-febrero-2018 a las 00:23

La definición de noCuadrado es incorrecta. Por ejemplo,

λ> noCuadrado 36 True

1
2

λ> noCuadrado 36
True

pero 36 es el cuadrado de 6.

Responder

import Data.List
import Graphics.Gnuplot.Simple

noCuadrados :: [Integer]
noCuadrados = [x | x <- [1..], (not . esCuadrado) x]
 
esCuadrado :: Integer -> Bool
esCuadrado n = elem n [x^2 | x <- [1..(floor.sqrt.fromIntegral) n]]
 
graficaNoCuadrados :: Integer -> IO ()
graficaNoCuadrados n =
  plotList [Key Nothing,
            Title ("No cuadrados hasta" ++ " " ++ show n)]
           (zipWith (-) noCuadrados [0..n])

import Data.List

import Graphics.Gnuplot.Simple

noCuadrados :: [Integer]

noCuadrados = [x | x <- [1..], (not . esCuadrado) x]

esCuadrado :: Integer -> Bool

esCuadrado n = elem n [x^2 | x <- [1..(floor.sqrt.fromIntegral) n]]

graficaNoCuadrados :: Integer -> IO ()

graficaNoCuadrados n =

plotList [Key Nothing,

Title ("No cuadrados hasta" ++ " " ++ show n)]

(zipWith (-) noCuadrados [0..n])

Responder

import Data.List
import Graphics.Gnuplot.Simple

noCuadrados :: [Integer]
noCuadrados =
  nub [x | x <- [2..]
         , (raizCuadradaEntera x) * (raizCuadradaEntera x)  /= x]

raizCuadradaEntera :: Integer -> Integer
raizCuadradaEntera n = truncate (sqrt (fromIntegral n))
 
graficaNoCuadrados :: Integer -> IO ()
graficaNoCuadrados n =
  plotList [ Title ("Grafica Diferencia con su posicion No Cuadrados hasta n, para n = " ++ show n)
           , Key Nothing]
           (genericTake n (noCuadradosPosicion))
 
noCuadradosPosicion :: [Integer]
noCuadradosPosicion = [sum [x,-y] | (x,y) <- zip noCuadrados ([1..])]

import Data.List

import Graphics.Gnuplot.Simple

noCuadrados :: [Integer]

noCuadrados =

nub [x | x <- [2..]

, (raizCuadradaEntera x) * (raizCuadradaEntera x) /= x]

raizCuadradaEntera :: Integer -> Integer

raizCuadradaEntera n = truncate (sqrt (fromIntegral n))

graficaNoCuadrados :: Integer -> IO ()

graficaNoCuadrados n =

plotList [ Title ("Grafica Diferencia con su posicion No Cuadrados hasta n, para n = " ++ show n)

, Key Nothing]

(genericTake n (noCuadradosPosicion))

noCuadradosPosicion :: [Integer]

noCuadradosPosicion = [sum [x,-y] | (x,y) <- zip noCuadrados ([1..])]

Responder

import Data.List
import Graphics.Gnuplot.Simple
import Control.Applicative

noCuadrados :: [Integer]
noCuadrados = concat $ (zipWith entre <*> tail) cuadrados
  where
    cuadrados = [n * n | n <- [1..]]
    entre n m = [n+1 .. m-1]

-- Eficiencia

-- λ> take 25 noCuadrados
-- [2,3,5,6,7,8,10,11,12,13,14,15,17,18,19,20,21,22,23,24,26,27,28,29,30]
-- (0.01 secs, 133,008 bytes)

-- λ>  noCuadrados !! 258
-- 275
-- (0.04 secs, 124,344 bytes)

-- λ>  genericLength(show(noCuadrados !! 25^378-8))
-- 564
-- (0.02 secs, 149,688 bytes)

-- λ> noCuadrados !! 25^375 
--18223642531500045957913062965842663289344226711176211217138728315956858997202735631855641013326051126299348620788459324466169779359523403583687117539543481521788500874368310345411295517397336946998392384488924758661435432752827225578319391010864658319515124984498559401119538137465219068290309136796140672041881111525138814440121251942415868896909814490207920964720866917897099320294765960247930646490972072999326789370646562620822064722399331168918177792691758947026943449336071997249237160839565061291805061750351081513165450091830699737779257463027398998751
-- (0.03 secs, 565,608 bytes)


-- Gráfica

graficaNoCuadrados :: Integer -> IO()
graficaNoCuadrados n  = plotList []  (zipWith (-) (genericTake n noCuadrados) [0..])


-- Propiedad
prop_noCuadrados :: Integer -> Property
prop_noCuadrados n = n>0 ==> noCuadrados !! (t-1) ==  (n + (floor (1/2 + sqrt(fromIntegral n))))
  where t = fromIntegral n

-- λ> quickCheck prop_noCuadrados
-- +++ OK, passed 100 tests.
-- (0.06 secs, 3,785,792 bytes)

import Data.List

import Graphics.Gnuplot.Simple

import Control.Applicative

noCuadrados :: [Integer]

noCuadrados = concat $ (zipWith entre <*> tail) cuadrados

where

cuadrados = [n * n | n <- [1..]]

entre n m = [n+1 .. m-1]

-- Eficiencia

-- λ> take 25 noCuadrados

-- [2,3,5,6,7,8,10,11,12,13,14,15,17,18,19,20,21,22,23,24,26,27,28,29,30]

-- (0.01 secs, 133,008 bytes)

-- λ> noCuadrados !! 258

-- 275

-- (0.04 secs, 124,344 bytes)

-- λ> genericLength(show(noCuadrados !! 25^378-8))

-- 564

-- (0.02 secs, 149,688 bytes)

-- λ> noCuadrados !! 25^375

--18223642531500045957913062965842663289344226711176211217138728315956858997202735631855641013326051126299348620788459324466169779359523403583687117539543481521788500874368310345411295517397336946998392384488924758661435432752827225578319391010864658319515124984498559401119538137465219068290309136796140672041881111525138814440121251942415868896909814490207920964720866917897099320294765960247930646490972072999326789370646562620822064722399331168918177792691758947026943449336071997249237160839565061291805061750351081513165450091830699737779257463027398998751

-- (0.03 secs, 565,608 bytes)

-- Gráfica

graficaNoCuadrados :: Integer -> IO()

graficaNoCuadrados n = plotList [] (zipWith (-) (genericTake n noCuadrados) [0..])

-- Propiedad

prop_noCuadrados :: Integer -> Property

prop_noCuadrados n = n>0 ==> noCuadrados !! (t-1) == (n + (floor (1/2 + sqrt(fromIntegral n))))

where t = fromIntegral n

-- λ> quickCheck prop_noCuadrados

-- +++ OK, passed 100 tests.

-- (0.06 secs, 3,785,792 bytes)

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