{"id":5893,"date":"2020-05-14T06:14:38","date_gmt":"2020-05-14T04:14:38","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=5893"},"modified":"2020-05-21T06:25:40","modified_gmt":"2020-05-21T04:25:40","slug":"cadenas-de-divisores","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/cadenas-de-divisores\/","title":{"rendered":"Cadenas de divisores"},"content":{"rendered":"<p>Una <strong>cadena de divisores<\/strong> de un n\u00famero n es una lista donde cada elemento es un divisor de su siguiente elemento en la lista. Por ejemplo, las cadenas de divisores de 12 son [2,4,12], [2,6,12], [2,12], [3,6,12], [3,12], [4,12], [6,12] y [12].<\/p>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   cadenasDivisores :: Int -> [[Int]]\n<\/pre>\n<p>tal que (cadenasDivisores n) es la lista de las cadenas de divisores de n. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   \u03bb> cadenasDivisores 12\n   [[2,4,12],[2,6,12],[2,12],[3,6,12],[3,12],[4,12],[6,12],[12]]\n   \u03bb> length (cadenaDivisores 48)\n   48\n   \u03bb> length (cadenaDivisores 120)\n   132\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (sort)\nimport Data.Numbers.Primes (isPrime)\n\n-- 1\u00aa definici\u00f3n\n-- =============\n\ncadenasDivisores :: Int -> [[Int]]\ncadenasDivisores n = sort (extiendeLista [[n]])\n    where extiendeLista []           = []\n          extiendeLista ((1:xs):yss) = xs : extiendeLista yss\n          extiendeLista ((x:xs):yss) =\n              extiendeLista ([y:x:xs | y <- divisores x] ++ yss)\n\n-- (divisores x) es la lista decreciente de los divisores de x distintos\n-- de x. Por ejemplo,\n--    divisores 12  ==  [6,4,3,2,1]\ndivisores :: Int -> [Int]\ndivisores x = \n    [y | y <- [a,a-1..1], x `mod` y == 0]\n    where a = x `div` 2\n\n-- 2\u00aa definici\u00f3n\n-- =============\n\ncadenasDivisores2 :: Int -> [[Int]]\ncadenasDivisores2 = sort . aux\n    where aux 1 = [[]]\n          aux n = [xs ++ [n] | xs <- concatMap aux (divisores n)]\n\n-- 3\u00aa definici\u00f3n\n-- =============\n\ncadenasDivisores3 :: Int -> [[Int]]\ncadenasDivisores3 = sort . map reverse . aux\n    where aux 1 = [[]]\n          aux n = map (n:) (concatMap aux (divisores3 n))\n\n-- (divisores3 x) es la lista creciente de los divisores de x distintos\n-- de x. Por ejemplo,\n--    divisores3 12  ==  [1,2,3,4,6]\ndivisores3 :: Int -> [Int]\ndivisores3 x = \n    [y | y <- [1..a], x `mod` y == 0]\n    where a = x `div` 2\n\n-- 1\u00aa definici\u00f3n de nCadenasDivisores\n-- ==================================\n\nnCadenasDivisores1 :: Int -> Int\nnCadenasDivisores1 = length . cadenasDivisores\n\n-- 2\u00aa definici\u00f3n de nCadenasDivisores\n-- ==================================\n\nnCadenasDivisores2 :: Int -> Int\nnCadenasDivisores2 1 = 1\nnCadenasDivisores2 n = \n    sum [nCadenasDivisores2 x | x <- divisores n]\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Una cadena de divisores de un n\u00famero n es una lista donde cada elemento es un divisor de su siguiente elemento en la lista. Por ejemplo, las cadenas de divisores de 12 son [2,4,12], [2,6,12], [2,12], [3,6,12], [3,12], [4,12], [6,12] y [12]. Definir la funci\u00f3n cadenasDivisores :: Int -> [[Int]] tal que (cadenasDivisores n) es&#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":[4],"tags":[8,58,30,28,10,89,11,6,14,40],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5893"}],"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=5893"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5893\/revisions"}],"predecessor-version":[{"id":5915,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5893\/revisions\/5915"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=5893"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=5893"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=5893"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}