{"id":5992,"date":"2021-01-20T06:03:41","date_gmt":"2021-01-20T04:03:41","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=5992"},"modified":"2021-01-27T08:35:56","modified_gmt":"2021-01-27T06:35:56","slug":"numeros-en-potencias-de-dos","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/numeros-en-potencias-de-dos\/","title":{"rendered":"N\u00fameros en potencias de dos"},"content":{"rendered":"<p>Las potencias de dos son<\/p>\n<pre lang=\"text\">\n   1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,...\n<\/pre>\n<p>Se observa que la primera potencia de dos que contiene al 638 es la 14 ya que 2^14 = 16384.<\/p>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   potenciasContenedoras :: Integer -> [Integer]\n<\/pre>\n<p>tal que (potenciasContenedoras x) es la lista de las potencias de 2 que contienen a x. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   \u03bb> head (potenciasContenedoras 638)\n   14\n   \u03bb> head (potenciasContenedoras 2021)\n   452\n   \u03bb> take 20 (potenciasContenedoras 4)\n   [2,6,10,11,12,14,18,19,20,22,25,26,27,28,30,31,32,33,34,35]\n   \u03bb> [head (potenciasContenedoras n) | n <- [0..20]]\n   [10,4,1,5,2,8,4,15,3,12,10,40,7,17,18,21,4,27,30,13,11]\n<\/pre>\n<p>Comprobar con QuickCheck si todos los n\u00fameros naturales est\u00e1n contenenidos en alguna potencia de 2.<\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (isInfixOf)\nimport Test.QuickCheck (Property, (==>), quickCheck)\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\npotenciasContenedoras :: Integer -> [Integer]\npotenciasContenedoras x =\n  [n | n <- [1..], x `estaContenidoEn` (2^n)]\n\n-- (estaContenidoEn x y) se verifica si el n\u00famero x est\u00e1 contenido en\n-- y. Por ejemplo,\n--    estaContenidoEn 23 42357  ==  True\n--    estaContenidoEn 23 42537  ==  False\nestaContenidoEn :: Integer -> Integer -> Bool\nestaContenidoEn x y = show x `isInfixOf` show y\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\npotenciasContenedoras2 :: Integer -> [Integer]\npotenciasContenedoras2 x =\n  [n | (n,y) <- zip [0..] potenciasDeDos, x `estaContenidoEn` y]\n\n-- potenciasDeDos es la lista de las potencias de dos. Por ejemplo,\n--    \u03bb> take 12 potenciasDeDos\n--    [1,2,4,8,16,32,64,128,256,512,1024,2048]\npotenciasDeDos :: [Integer]\npotenciasDeDos = iterate (*2) 1\n\n-- Propiedad\n-- =========\n\n-- La propiedad es\nprop_potenciasContenedoras :: Integer -> Property\nprop_potenciasContenedoras n =\n  n > 0 ==> not (null (potenciasContenedoras n))\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_potenciasContenedoras\n--    +++ OK, passed 100 tests.\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Las potencias de dos son 1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,&#8230; Se observa que la primera potencia de dos que contiene al 638 es la 14 ya que 2^14 = 16384. Definir la funci\u00f3n potenciasContenedoras :: Integer -> [Integer] tal que (potenciasContenedoras x) es la lista de las potencias de 2 que contienen a x. Por ejemplo, \u03bb> head (potenciasContenedoras&#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":[5],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5992"}],"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=5992"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5992\/revisions"}],"predecessor-version":[{"id":6030,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5992\/revisions\/6030"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=5992"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=5992"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=5992"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}