{"id":5336,"date":"2020-01-07T05:30:04","date_gmt":"2020-01-07T03:30:04","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=5336"},"modified":"2020-01-14T21:32:27","modified_gmt":"2020-01-14T19:32:27","slug":"la-mayor-potencia-de-dos-no-es-divisor","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/la-mayor-potencia-de-dos-no-es-divisor\/","title":{"rendered":"La mayor potencia de dos no es divisor"},"content":{"rendered":"<p>Para cada n\u00famero entero positivo n, se define el conjunto<\/p>\n<pre lang=\"text\">\n   S(n) = {1, 2, 3, ..., n}\n<\/pre>\n<p>de los n\u00fameros desde el 1 hasta n.<\/p>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   mayorPotenciaDeDosEnS :: Integer -> Integer\n<\/pre>\n<p>tal que (mayorPotenciaDeDosEnS n) es la mayor potencia de 2 en S(n). Por ejemplo,<\/p>\n<pre lang=\"text\">\n   mayorPotenciaDeDosEnS 3  ==  2\n   mayorPotenciaDeDosEnS 4  ==  4\n<\/pre>\n<p>Comprobar con QuickCheck que la mayor potencia de 2 en S(n) no divide a ning\u00fan otro elemento de S(n).<\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (delete)\nimport Test.QuickCheck\n\nmayorPotenciaDeDosEnS :: Integer -> Integer\nmayorPotenciaDeDosEnS n =\n  last (takeWhile (<=n) potenciasDeDos)\n\n-- potenciasDeDos es la lista de las potencias de 2. Por ejemplo,\n--    take 5 potenciasDeDos  ==  [1,2,4,8,16]\npotenciasDeDos :: [Integer]\npotenciasDeDos = iterate (*2) 1\n\n-- La propiedad es\nprop_mayorPotenciaDeDosEnS :: Integer -> Property\nprop_mayorPotenciaDeDosEnS n =\n  n > 0 ==> all (x `noDivide`) (delete x [1..n])\n  where x = mayorPotenciaDeDosEnS n\n        x `noDivide` y = y `mod` x \/= 0\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_mayorPotenciaDeDosEnS\n--    +++ OK, passed 100 tests.\n<\/pre>\n<h4>Referencia<\/h4>\n<ul>\n<li>Basado en <a href=\"http:\/\/bit.ly\/3784bRe\">Greatest power of two not divisor<\/a> de ProofWiki.<\/li>\n<\/ul>\n<h4>Pensamiento<\/h4>\n<blockquote><p>\n\u00a1S\u00f3lo tu figura,<br \/>\ncomo una centella blanca,<br \/>\nen mi noche oscura.<\/p>\n<p>Antonio Machado\n<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>Para cada n\u00famero entero positivo n, se define el conjunto S(n) = {1, 2, 3, &#8230;, n} de los n\u00fameros desde el 1 hasta n. Definir la funci\u00f3n mayorPotenciaDeDosEnS :: Integer -> Integer tal que (mayorPotenciaDeDosEnS n) es la mayor potencia de 2 en S(n). Por ejemplo, mayorPotenciaDeDosEnS 3 == 2 mayorPotenciaDeDosEnS 4 == 4&#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":[7],"tags":[41,25,50,415,89,11,34,146],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5336"}],"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=5336"}],"version-history":[{"count":4,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5336\/revisions"}],"predecessor-version":[{"id":5395,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5336\/revisions\/5395"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=5336"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=5336"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=5336"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}