Sucesión de Cantor de números innombrables

Un número es innombrable si es divisible por 7 o alguno de sus dígitos es un 7. Un juego infantil consiste en contar saltándose los números innombrables:

   1 2 3 4 5 6 ( ) 8 9 10 11 12 13 ( ) 15 16 ( ) 18 ...

1	1 2 3 4 5 6 ( ) 8 9 10 11 12 13 ( ) 15 16 ( ) 18 ...

La sucesión de Cantor se obtiene llenando los huecos de la sucesión anterior:

  1 2 3 4 5 6 (1) 8 9 10 11 12 13 (2) 15 16 (3) 18 19 20 (4) 22 23
  24 25 26 (5) (6) 29 30 31 32 33 34 (1) 36 (8) 38 39 40 41  (9) 43
  44 45 46 (10) 48 (11) 50 51 52 53 54 55 (12) (13) 58 59 60 61 62
  (2) 64 65 66 (15) 68 69 (16) (3) (18) (19) (20) (4) (22) (23) (24)
  (25) 80 81 82 83 (26) 85 86 (5) 88 89 90 (6) 92 93 94 95 96 (29)
  (30) 99 100

1 2 3 4 5 6 (1) 8 9 10 11 12 13 (2) 15 16 (3) 18 19 20 (4) 22 23

24 25 26 (5) (6) 29 30 31 32 33 34 (1) 36 (8) 38 39 40 41 (9) 43

44 45 46 (10) 48 (11) 50 51 52 53 54 55 (12) (13) 58 59 60 61 62

(2) 64 65 66 (15) 68 69 (16) (3) (18) (19) (20) (4) (22) (23) (24)

(25) 80 81 82 83 (26) 85 86 (5) 88 89 90 (6) 92 93 94 95 96 (29)

(30) 99 100

Definir las funciones

   sucCantor        :: [Integer]
   graficaSucCantor :: Int -> IO ()

1 2	sucCantor :: [Integer] graficaSucCantor :: Int -> IO ()

tales que

sucCantor es la lista cuyos elementos son los términos de la sucesión de Cantor. Por ejemplo,

     λ> take 100 sucCantor
     [1,2,3,4,5,6, 1 ,8,9,10,11,12,13, 2, 15,16, 3, 18,19,20, 4,
      22,23,24,25,26, 5 , 6 ,29,30,31,32,33,34, 1 ,36 , 8 ,38,39,
      40,41, 9 ,43,44,45,46, 10 ,48, 11 ,50,51,52,53,54,55 , 12 ,
      13, 58,59,60,61,62, 2 ,64,65,66, 15 ,68,69, 16 , 3 , 18, 19,
      20, 4, 22, 23, 24 ,25 ,80,81,82,83, 26 ,85,86, 5 ,88,89,90,
      6, 92,93,94,95,96, 29, 30 ,99,100]
     λ> sucCantor2 !! (5+10^6)
     544480
     λ> sucCantor2 !! (6+10^6)
     266086

λ> take 100 sucCantor

[1,2,3,4,5,6, 1 ,8,9,10,11,12,13, 2, 15,16, 3, 18,19,20, 4,

22,23,24,25,26, 5 , 6 ,29,30,31,32,33,34, 1 ,36 , 8 ,38,39,

40,41, 9 ,43,44,45,46, 10 ,48, 11 ,50,51,52,53,54,55 , 12 ,

13, 58,59,60,61,62, 2 ,64,65,66, 15 ,68,69, 16 , 3 , 18, 19,

20, 4, 22, 23, 24 ,25 ,80,81,82,83, 26 ,85,86, 5 ,88,89,90,

6, 92,93,94,95,96, 29, 30 ,99,100]

λ> sucCantor2 !! (5+10^6)

544480

λ> sucCantor2 !! (6+10^6)

266086

(graficaSucCantor n) es la gráfica de los n primeros términos de la sucesión de Cantor. Por ejemplo, (graficaSucCantor 200) dibuja

Soluciones

import Graphics.Gnuplot.Simple

-- 1ª solución
-- ===========

sucCantor1 :: [Integer]
sucCantor1 = map fst $ scanl f (1,0) [2..]
  where f (a,i) x
          | esInnombrable x = (sucCantor1 !! i, i+1)
          | otherwise       = (x,i)

esInnombrable :: Integer -> Bool
esInnombrable x =
  rem x 7 == 0 || '7' `elem` show x

-- 2ª solución
-- ===========

sucCantor2 :: [Integer]
sucCantor2 = aux 0 1
  where aux i x
          | esInnombrable x = sucCantor2 !! i : aux (i+1) (x+1)
          | otherwise       = x : aux i (x+1) 

-- 3ª solución
-- ===========

sucCantor3 :: [Integer]
sucCantor3 = 1 : aux [2..] sucCantor3
  where aux [] _ = []
        aux (x:xs) a@(y:ys)
          | esInnombrable x = y : aux xs ys
          | otherwise       = x : aux xs a

-- Definición de graficaSucCantor
-- ========================================

graficaSucCantor :: Int -> IO ()
graficaSucCantor n =
  plotList [ Key Nothing
           , PNG ("Sucesion_de_Cantor_de_numeros_innombrables.png")
           ]
           (take n sucCantor3)

import Graphics.Gnuplot.Simple

-- 1ª solución

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

sucCantor1 :: [Integer]

sucCantor1 = map fst $ scanl f (1,0) [2..]

where f (a,i) x

| esInnombrable x = (sucCantor1 !! i, i+1)

| otherwise = (x,i)

esInnombrable :: Integer -> Bool

esInnombrable x =

rem x 7 == 0 || '7' `elem` show x

-- 2ª solución

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

sucCantor2 :: [Integer]

sucCantor2 = aux 0 1

where aux i x

| esInnombrable x = sucCantor2 !! i : aux (i+1) (x+1)

| otherwise = x : aux i (x+1)

-- 3ª solución

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

sucCantor3 :: [Integer]

sucCantor3 = 1 : aux [2..] sucCantor3

where aux [] _ = []

aux (x:xs) a@(y:ys)

| esInnombrable x = y : aux xs ys

| otherwise = x : aux xs a

-- Definición de graficaSucCantor

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

graficaSucCantor :: Int -> IO ()

graficaSucCantor n =

plotList [ Key Nothing

, PNG ("Sucesion_de_Cantor_de_numeros_innombrables.png")

]

(take n sucCantor3)

Pensamiento

Dices que nada se pierde
y acaso dices verdad;
pero todo lo perdemos
y todo nos perderá.

Antonio Machado

4 Comentarios

import Graphics.Gnuplot.Simple

esInnombrable :: Integer -> Bool
esInnombrable n = mod n 7 == 0 || elem '7' (show n)

sucCantor :: [Integer]
sucCantor = sucCantorAux 1 0
 where sucCantorAux x y | esInnombrable x =
                          (sucCantor !! y) : sucCantorAux (x+1) (y+1)
                        | otherwise = x : sucCantorAux (x+1) y

graficaSucCantor :: Int -> IO ()
graficaSucCantor n = 
  plotList [ Key Nothing
           , PNG "sucCantor.png"]
           (take n sucCantor)

import Graphics.Gnuplot.Simple

esInnombrable :: Integer -> Bool

esInnombrable n = mod n 7 == 0 || elem '7' (show n)

sucCantor :: [Integer]

sucCantor = sucCantorAux 1 0

where sucCantorAux x y | esInnombrable x =

(sucCantor !! y) : sucCantorAux (x+1) (y+1)

| otherwise = x : sucCantorAux (x+1) y

graficaSucCantor :: Int -> IO ()

graficaSucCantor n =

plotList [ Key Nothing

, PNG "sucCantor.png"]

(take n sucCantor)

Responder

sucCantor :: [Integer]
sucCantor = sucAux  [1..] [1..]
  where sucAux (x:xs) (y:ys) | innombrable x = y:sucAux xs ys
                             | otherwise = x:sucAux xs (y:ys)

innombrable :: Integer -> Bool
innombrable x = x`mod`7 == 0 || '7'`elem`show x

graficaSucCantor :: Int -> IO ()
graficaSucCantor n =
  plotList [ Key Nothing
           , PNG ("Sucesion_de_Cantor_de_numeros_innombrables.png") ]
            (take n sucCantor3)

sucCantor :: [Integer]

sucCantor = sucAux [1..] [1..]

where sucAux (x:xs) (y:ys) | innombrable x = y:sucAux xs ys

| otherwise = x:sucAux xs (y:ys)

innombrable :: Integer -> Bool

innombrable x = x`mod`7 == 0 || '7'`elem`show x

graficaSucCantor :: Int -> IO ()

graficaSucCantor n =

plotList [ Key Nothing

, PNG ("Sucesion_de_Cantor_de_numeros_innombrables.png") ]

(take n sucCantor3)

Responder

--No habia entendido muy bien el enunciado, pero ya lo he corregido
sucCantor :: [Integer]
sucCantor = sucAux  [1..] [1..] 0
  where sucAux (x:xs) (y:ys) n | innombrable x && not ( innombrable y) = y:sucAux xs ys n
                                              | innombrable x && innombrable y = (sucCantor!!n):sucAux xs ys (n+1)
                                              | otherwise = x:sucAux xs (y:ys) n 
innombrable :: Integer -> Bool
innombrable x = x`mod`7 == 0 || '7'`elem`show x

--No habia entendido muy bien el enunciado, pero ya lo he corregido

sucCantor :: [Integer]

sucCantor = sucAux [1..] [1..] 0

where sucAux (x:xs) (y:ys) n | innombrable x && not ( innombrable y) = y:sucAux xs ys n

| innombrable x && innombrable y = (sucCantor!!n):sucAux xs ys (n+1)

| otherwise = x:sucAux xs (y:ys) n

innombrable :: Integer -> Bool

innombrable x = x`mod`7 == 0 || '7'`elem`show x

Responder

sucCantor        :: [Integer]
sucCantor = aux 1 0
  where aux n a | mod n 7 == 0 || elem '7' (show n ) =  (sucCantor !! a) : aux (n+1) (a+1)
                | otherwise = n : aux (n+1) a

graficaSucCantor :: Int -> IO ()
graficaSucCantor n = plotList [] (take n sucCantor)

sucCantor :: [Integer]

sucCantor = aux 1 0

where aux n a | mod n 7 == 0 || elem '7' (show n ) = (sucCantor !! a) : aux (n+1) (a+1)

| otherwise = n : aux (n+1) a

graficaSucCantor :: Int -> IO ()

graficaSucCantor n = plotList [] (take n sucCantor)

Responder

Sucesión de Cantor de números innombrables

Soluciones

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