{"id":6278,"date":"2021-04-16T06:00:56","date_gmt":"2021-04-16T04:00:56","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=6278"},"modified":"2021-04-23T07:57:12","modified_gmt":"2021-04-23T05:57:12","slug":"sucesion-de-cubos-perfectos","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/sucesion-de-cubos-perfectos\/","title":{"rendered":"Sucesi\u00f3n de cubos perfectos"},"content":{"rendered":"<p>Definir la lista<\/p>\n<pre lang=\"text\">\n   sucesion :: [Integer]\n<\/pre>\n<p>cuyos elementos son los t\u00e9rminos de la sucesi\u00f3n<\/p>\n<pre lang=\"text\">\n   107811\/3, 110778111\/3, 111077781111\/3, 111107777811111\/3, ...\n<\/pre>\n<p>Por ejemplo,<\/p>\n<pre lang=\"text\">\n   \u03bb> take 5 sucesion\n   [35937,36926037,37025927037,37035925937037,37036925926037037]\n   \u03bb> length (show (sucesion2 !! (3*10^6)))\n   9000005\n<\/pre>\n<p>Comprobar con QuickCheck que todos los t\u00e9rminos de la sucesi\u00f3n son cubos perfectos.<\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Test.QuickCheck\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nsucesion :: [Integer]\nsucesion = map termino [1..]\n\n-- (termino n) es el t\u00e9rmino n-\u00e9simo de la sucesi\u00f3n por ejemplo,\n--    termino 1  ==  35937\n--    termino 2  ==  36926037\n--    termino 3  ==  37025927037\ntermino :: Int -> Integer\ntermino n = numerador n `div` 3\n\n\n-- (numerador n) es el numerador del t\u00e9rmino n-\u00e9simo de la sucesi\u00f3n por\n-- ejemplo,\n--    \u03bb> mapM_ print (map numerador [1..5])\n--    107811\n--    110778111\n--    111077781111\n--    111107777811111\n--    111110777778111111\nnumerador :: Int -> Integer\nnumerador n =\n  read (replicate n '1' ++ \"0\" ++ replicate n '7' ++ \"81\" ++ replicate n '1')\n\n-- Propiedad\n-- =========\n\n-- La propiedad es\nprop_sucesion :: Int -> Property\nprop_sucesion n =\n  n >= 0 ==>\n  esCubo (sucesion !! n)\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_sucesion\n--    +++ OK, passed 100 tests.\n\n-- (esCubo x) se verifica si x es un cubo perfecto. Por ejemplo,\n--    esCubo 27  ==  True\n--    esCubo 28  ==  False\nesCubo :: Integer -> Bool\nesCubo x =\n  (raizCubicaEntera x)^3 == x\n\n-- (raizCubicaEntera x) es el mayor entero cuyo cubo es menor o igual\n-- que x. Por ejemplo,\n--    raizCubicaEntera 26  ==  2\n--    raizCubicaEntera 27  ==  3\n--    raizCubicaEntera 28  ==  3\nraizCubicaEntera :: Integer -> Integer\nraizCubicaEntera x = aux (1,x)\n    where aux (a,b) | d == x    = c\n                    | c == a    = c\n                    | d < x     = aux (c,b)\n                    | otherwise = aux (a,c)\n              where c = (a+b) `div` 2\n                    d = c^3\n\n-- Otra forma es observando los cubos de los t\u00e9rminos de la sucesi\u00f3n\n--    \u03bb> map raizCubicaEntera (take 7 sucesion)\n--    [33,333,3333,33333,333333,3333333,33333333]\n\n-- La propiedad es\nprop_sucesion2 :: Int -> Property\nprop_sucesion2 n =\n  n >= 0 ==>\n  sucesion !! n == ((10^(n+2)-1) `div` 3)^3\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_sucesion2\n--    +++ OK, passed 100 tests.\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\n-- Basada en la propiedad anterior.\n\nsucesion2 :: [Integer]\nsucesion2 =\n  [((10^n-1) `div` 3)^3 | n <- [2..]]\n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_equiv :: Int -> Property\nprop_equiv n =\n  n >= 0 ==>\n  sucesion !! n == sucesion2 !! n\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_equiv\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> length (show (sucesion !! (3*10^6)))\n--    9000005\n--    (6.34 secs, 5,161,177,808 bytes)\n--    \u03bb> length (show (sucesion2 !! (3*10^6)))\n--    9000005\n--    (2.72 secs, 1,124,065,328 bytes)\n<\/pre>\n<h4>Nuevas soluciones<\/h4>\n<ul>\n<li>En los comentarios se pueden escribir nuevas soluciones.\n<li>El c\u00f3digo se debe escribir entre una l\u00ednea con &#60;pre lang=&quot;haskell&quot;&#62; y otra con &#60;\/pre&#62;\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Definir la lista sucesion :: [Integer] cuyos elementos son los t\u00e9rminos de la sucesi\u00f3n 107811\/3, 110778111\/3, 111077781111\/3, 111107777811111\/3, &#8230; Por ejemplo, \u03bb> take 5 sucesion [35937,36926037,37025927037,37035925937037,37036925926037037] \u03bb> length (show (sucesion2 !! (3*10^6))) 9000005 Comprobar con QuickCheck que todos los t\u00e9rminos de la sucesi\u00f3n son cubos perfectos. Soluciones import Test.QuickCheck &#8212; 1\u00aa soluci\u00f3n &#8212; =========== sucesion&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[2],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6278"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/comments?post=6278"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6278\/revisions"}],"predecessor-version":[{"id":6328,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6278\/revisions\/6328"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=6278"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=6278"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=6278"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}