{"id":7014,"date":"2022-05-12T17:34:06","date_gmt":"2022-05-12T15:34:06","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=7014"},"modified":"2022-05-12T17:41:44","modified_gmt":"2022-05-12T15:41:44","slug":"conjunto-de-divisores","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/conjunto-de-divisores\/","title":{"rendered":"Conjunto de divisores"},"content":{"rendered":"<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   divisores :: Integer -> [Integer]\n<\/pre>\n<p>tal que <code>(divisores x)<\/code> es el conjunto de divisores de <code>x<\/code>. Por ejemplo,<\/p>\n<pre lang=\"text\">\n  divisores 30  ==  [1,2,3,5,6,10,15,30]\n  length (divisores (product [1..10]))  ==  270\n  length (divisores (product [1..25]))  ==  340032\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (group, inits, nub, sort, subsequences)\nimport Data.Numbers.Primes (primeFactors)\nimport Test.QuickCheck\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\ndivisores1 :: Integer -> [Integer]\ndivisores1 n = [x | x <- [1..n], n `rem` x == 0]\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\ndivisores2 :: Integer -> [Integer]\ndivisores2 n = filter ((== 0) . mod n) [1..n]\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\ndivisores3 :: Integer -> [Integer]\ndivisores3 =\n  nub . sort . map product . subsequences . primeFactors\n\n-- 4\u00aa soluci\u00f3n\n-- ===========\n\ndivisores4 :: Integer -> [Integer]\ndivisores4 =\n  sort\n  . map (product . concat)\n  . productoCartesiano\n  . map inits\n  . group\n  . primeFactors\n\n-- (productoCartesiano xss) es el producto cartesiano de los conjuntos\n-- xss. Por ejemplo,\n--    \u03bb> productoCartesiano [[1,3],[2,5],[6,4]]\n--    [[1,2,6],[1,2,4],[1,5,6],[1,5,4],[3,2,6],[3,2,4],[3,5,6],[3,5,4]]\nproductoCartesiano :: [[a]] -> [[a]]\nproductoCartesiano []       = [[]]\nproductoCartesiano (xs:xss) =\n  [x:ys | x <- xs, ys <- productoCartesiano xss]\n\n-- 5\u00aa soluci\u00f3n\n-- ===========\n\ndivisores5 :: Integer -> [Integer]\ndivisores5 = sort\n           . map (product . concat)\n           . sequence\n           . map inits\n           . group\n           . primeFactors\n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_divisores :: Positive Integer -> Bool\nprop_divisores (Positive n) =\n  all (== divisores1 n)\n      [ divisores2 n\n      , divisores3 n\n      , divisores4 n\n      , divisores5 n\n      ]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_divisores\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de la eficiencia\n-- ============================\n\n--    \u03bb> length (divisores (product [1..11]))\n--    540\n--    (12.51 secs, 7,983,499,736 bytes)\n--    \u03bb> length (divisores2 (product [1..11]))\n--    540\n--    (4.81 secs, 4,790,146,656 bytes)\n--    \u03bb> length (divisores3 (product [1..11]))\n--    540\n--    (0.10 secs, 107,339,848 bytes)\n--    \u03bb> length (divisores4 (product [1..11]))\n--    540\n--    (0.02 secs, 1,702,616 bytes)\n--    \u03bb> length (divisores5 (product [1..11]))\n--    540\n--    (0.02 secs, 1,205,824 bytes)\n--\n--    \u03bb> length (divisores3 (product [1..14]))\n--    2592\n--    (7.89 secs, 9,378,454,912 bytes)\n--    \u03bb> length (divisores4 (product [1..14]))\n--    2592\n--    (0.03 secs, 9,426,528 bytes)\n--    \u03bb> length (divisores5 (product [1..14]))\n--    2592\n--    (0.02 secs, 6,636,608 bytes)\n--    \n--    \u03bb> length (divisores4 (product [1..25]))\n--    340032\n--    (1.65 secs, 2,055,558,208 bytes)\n--    \u03bb> length (divisores5 (product [1..25]))\n--    340032\n--    (0.88 secs, 1,532,515,304 bytes)\n<\/pre>\n<p>El c\u00f3digo se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Exercitium\/blob\/main\/src\/Conjunto_de_divisores.hs\">GitHub<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Definir la funci\u00f3n divisores :: Integer -> [Integer] tal que (divisores x) es el conjunto de divisores de x. Por ejemplo, divisores 30 == [1,2,3,5,6,10,15,30] length (divisores (product [1..10])) == 270 length (divisores (product [1..25])) == 340032 Soluciones import Data.List (group, inits, nub, sort, subsequences) import Data.Numbers.Primes (primeFactors) import Test.QuickCheck &#8212; 1\u00aa soluci\u00f3n &#8212; ===========&#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":[41,8,12,498,501,38,13,74,10,89,24,11,247,157,6,31,14,88,521,146],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7014"}],"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=7014"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7014\/revisions"}],"predecessor-version":[{"id":7016,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7014\/revisions\/7016"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=7014"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=7014"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=7014"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}