{"id":1971,"date":"2016-01-11T06:00:35","date_gmt":"2016-01-11T04:00:35","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=1971"},"modified":"2016-01-18T08:52:06","modified_gmt":"2016-01-18T06:52:06","slug":"parte-libre-de-cuadrados-y-parte-cuadrada-de-un-numero","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/parte-libre-de-cuadrados-y-parte-cuadrada-de-un-numero\/","title":{"rendered":"Parte libre de cuadrados y parte cuadrada de un n\u00famero"},"content":{"rendered":"<p>La <strong>parte libre de cuadrados<\/strong> de un n\u00famero n es el producto de todos sus divisores primos con exponente impar en la factorizaci\u00f3n prima de n. Por ejemplo, la parte libre de cuadrados de 360 es 10 ya que 360 = 2\u00b33\u00b25 y 2.5 = 10; adem\u00e1s, 360 = 10.6\u00b2<\/p>\n<p>La <strong>parte cuadrada<\/strong> de un n\u00famero n es el mayor n\u00famero cuadrado que divide a n. Por ejemplo, la parte cuadrada de 360 es 6.<\/p>\n<p>Definir las funciones<\/p>\n<pre lang=\"text\">\n   parteLibre    :: Integer -> Integer\n   parteCuadrada :: Integer -> Integer\n<\/pre>\n<p>tales que<\/p>\n<ul>\n<li>(parteLibre x) es la parte libre de x. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     parteLibre 360                 ==  10\n     parteLibre 1800                ==  2\n     [parteLibre n | n <- [1..14]]  ==  [1,2,3,1,5,6,7,2,1,10,11,3,13,14]\n<\/pre>\n<ul>\n<li>(parteCuadrada x) es la parte cuadrada de x. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     parteCuadrada 360                 ==  36\n     parteCuadrada 1800                ==  900\n     [parteCuadrada n | n <- [1..14]]  ==  [1,1,1,4,1,1,1,4,9,1,1,4,1,1]\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (group, genericLength)\nimport Data.Numbers.Primes (primeFactors)\n\nparteLibre :: Integer -> Integer\nparteLibre n = product [p | (p,e) <- factorizacion n, odd e]\n\n-- (factorizacion n) es la factorizaci\u00f3n de n. Por ejemplo, \n--    factorizacion  600  ==  [(2,3),(3,1),(5,2)]\n--    factorizacion 1400  ==  [(2,3),(5,2),(7,1)]\nfactorizacion :: Integer -> [(Integer,Integer)]\nfactorizacion n =\n    [(head xs,genericLength xs) | xs <- group (primeFactors n)]\n\nparteCuadrada :: Integer -> Integer\nparteCuadrada n = n `div` parteLibre n\n<\/pre>\n<h4>Referencias<\/h4>\n<ul>\n<li>N.J.A. Sloane, <a href=\"http:\/\/bit.ly\/1SbieuU\">Sequence A008833<\/a> en OEIS. <\/li>\n<li>N.J.A. Sloane, <a href=\"http:\/\/bit.ly\/1Sbifz8\">Sequence A007913<\/a> en OEIS. <\/li>\n<li>E.W. Weisstein, <a href=\"http:\/\/bit.ly\/1Sbi7Qb\">Square part<\/a> en MathWorld. <\/li>\n<li>E.W. Weisstein, <a href=\"http:\/\/bit.ly\/1SbhZ3h\">Squarefree part<\/a> en MathWorld. <\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>La parte libre de cuadrados de un n\u00famero n es el producto de todos sus divisores primos con exponente impar en la factorizaci\u00f3n prima de n. Por ejemplo, la parte libre de cuadrados de 360 es 10 ya que 360 = 2\u00b33\u00b25 y 2.5 = 10; adem\u00e1s, 360 = 10.6\u00b2 La parte cuadrada de un&#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":[8,30,258,13,71,92,247,157],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/1971"}],"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=1971"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/1971\/revisions"}],"predecessor-version":[{"id":2007,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/1971\/revisions\/2007"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=1971"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=1971"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=1971"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}