{"id":7030,"date":"2022-05-19T16:17:50","date_gmt":"2022-05-19T14:17:50","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=7030"},"modified":"2022-05-19T16:17:50","modified_gmt":"2022-05-19T14:17:50","slug":"parejas-de-numeros-y-divisores","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/parejas-de-numeros-y-divisores\/","title":{"rendered":"Parejas de n\u00fameros y divisores"},"content":{"rendered":"<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   divisoresHasta :: Int -> [(Int,Int)]\n<\/pre>\n<p>tal que <code>(divisoresHasta n)<\/code> es la lista de los pares <code>(a,b)<\/code> tales que <code>a<\/code> es un n\u00famero entre 2 y <code>n<\/code> y <code>b<\/code> es un divisor propio de <code>a<\/code>. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   \u03bb> divisoresHasta 6\n   [(2,1),(3,1),(4,1),(5,1),(6,1),(4,2),(6,2),(6,3)]\n   \u03bb> divisoresHasta 8\n   [(2,1),(3,1),(4,1),(5,1),(6,1),(7,1),(8,1),(4,2),(6,2),(8,2),(6,3),(8,4)]\n   \u03bb> length (divisoresHasta 1234567)\n   16272448\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (sort, sortBy)\nimport Data.Ord (comparing)\nimport Data.Tuple (swap)\nimport Test.QuickCheck\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\ndivisoresHasta1 :: Int -> [(Int,Int)]\ndivisoresHasta1 n =\n  ordena (concat [[(a,b) | b <- divisoresPropios a] | a <- [2..n]])\n  where ordena ps = [intercambia p | p <- sort (map intercambia ps)]\n        intercambia (x,y) = (y,x)\n\ndivisoresPropios :: Int -> [Int]\ndivisoresPropios n = [x | x <- [1..n `div` 2], n `mod` x == 0]\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\ndivisoresHasta2 :: Int -> [(Int,Int)]\ndivisoresHasta2 n =\n  ordena (concat [[(a,b) | b <- divisoresPropios a] | a <- [2..n]])\n  where ordena           = sortBy comp\n        comp (x,y) (u,v) = compare (y,x) (v,u)\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\ndivisoresHasta3 :: Int -> [(Int,Int)]\ndivisoresHasta3 n =\n  ordena (concat [[(a,b) | b <- divisoresPropios a] | a <- [2..n]])\n  where ordena = sortBy (comparing swap)\n\n-- 4\u00aa soluci\u00f3n\n-- ===========\n\ndivisoresHasta4 :: Int -> [(Int,Int)]\ndivisoresHasta4 n =\n  [(a,b) | b <- [1..n `div` 2], a <- [2*b, 3*b..n]]\n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_divisoresHasta :: Positive Int -> Bool\nprop_divisoresHasta (Positive n) =\n  all (== divisoresHasta1 n)\n      [ divisoresHasta2 n\n      , divisoresHasta3 n\n      , divisoresHasta4 n\n      ]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_divisoresHasta\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> length (divisoresHasta1 4000)\n--    29805\n--    (1.78 secs, 843,584,472 bytes)\n--    \u03bb> length (divisoresHasta2 4000)\n--    29805\n--    (1.78 secs, 879,421,056 bytes)\n--    \u03bb> length (divisoresHasta3 4000)\n--    29805\n--    (1.70 secs, 876,475,584 bytes)\n--    \u03bb> length (divisoresHasta4 4000)\n--    29805\n--    (0.02 secs, 6,332,016 bytes)\n<\/pre>\n<p>El c\u00f3digo se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Exercitium\/blob\/main\/src\/Parejas_de_numeros_y_divisores.hs\">GitHub<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Definir la funci\u00f3n divisoresHasta :: Int -> [(Int,Int)] tal que (divisoresHasta n) es la lista de los pares (a,b) tales que a es un n\u00famero entre 2 y n y b es un divisor propio de a. Por ejemplo, \u03bb> divisoresHasta 6 [(2,1),(3,1),(4,1),(5,1),(6,1),(4,2),(6,2),(6,3)] \u03bb> divisoresHasta 8 [(2,1),(3,1),(4,1),(5,1),(6,1),(7,1),(8,1),(4,2),(6,2),(8,2),(6,3),(8,4)] \u03bb> length (divisoresHasta 1234567) 16272448 Soluciones import Data.List&#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":[521],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7030"}],"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=7030"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7030\/revisions"}],"predecessor-version":[{"id":7031,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7030\/revisions\/7031"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=7030"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=7030"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=7030"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}