{"id":7018,"date":"2022-05-13T13:02:11","date_gmt":"2022-05-13T11:02:11","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=7018"},"modified":"2022-05-13T13:02:11","modified_gmt":"2022-05-13T11:02:11","slug":"numero-de-divisores","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/numero-de-divisores\/","title":{"rendered":"N\u00famero de divisores"},"content":{"rendered":"<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   numeroDivisores :: Integer -> Integer\n<\/pre>\n<p>tal que <code>(numeroDivisores x)<\/code> es el n\u00famero de divisores de <code>x<\/code>. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   numeroDivisores 12  ==  6\n   numeroDivisores 25  ==  3\n   length (show (numeroDivisores (product [1..3*10^4])))  ==  1948\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (genericLength, group, inits)\nimport Data.Numbers.Primes (primeFactors)\nimport Test.QuickCheck\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nnumeroDivisores1 :: Integer -> Integer\nnumeroDivisores1 x =\n  genericLength [y | y <- [1..x], x `mod` y == 0]\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nnumeroDivisores2 :: Integer -> Integer\nnumeroDivisores2 1 = 1\nnumeroDivisores2 x\n  | esCuadrado x = 2 * genericLength [y | y <- [1..raizEntera x], x `mod` y == 0] - 1\n  | otherwise    = 2 * genericLength [y | y <- [1..raizEntera x], x `mod` y == 0]\n\n-- (raizEntera x) es el mayor n\u00famero entero cuyo cuadrado es menor o\n-- igual que x. Por ejemplo,\n--    raizEntera 3  ==  1\n--    raizEntera 4  ==  2\n--    raizEntera 5  ==  2\n--    raizEntera 8  ==  2\n--    raizEntera 9  ==  3\nraizEntera :: Integer -> Integer\nraizEntera x = floor (sqrt (fromInteger x))\n\n-- (esCuadrado x) se verifica si x es un cuadrado perfecto. Por ejemplo,\n--    esCuadrado 9  ==  True\n--    esCuadrado 7  ==  False\nesCuadrado :: Integer -> Bool\nesCuadrado x =\n  x == (raizEntera x)^2\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\nnumeroDivisores3 :: Integer -> Integer\nnumeroDivisores3 =\n  genericLength . divisores\n\n-- (divisores x) es la lista de los divisores de x. Por ejemplo,\n--    divisores 12  ==  [1,3,2,6,4,12]\n--    divisores 25  ==  [1,5,25]\ndivisores :: Integer -> [Integer]\ndivisores = 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-- 4\u00aa soluci\u00f3n\n-- ===========\n\nnumeroDivisores4 :: Integer -> Integer\nnumeroDivisores4 = genericLength\n                 . map (product . concat)\n                 . sequence\n                 . map inits\n                 . group\n                 . primeFactors\n\n-- 5\u00aa soluci\u00f3n\n-- ===========\n\nnumeroDivisores5 :: Integer -> Integer\nnumeroDivisores5 =\n  product . map ((+1) . genericLength) . group . primeFactors\n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_numeroDivisores :: Positive Integer -> Bool\nprop_numeroDivisores (Positive x) =\n  all (== numeroDivisores1 x)\n      [ numeroDivisores2 x\n      , numeroDivisores3 x\n      , numeroDivisores4 x\n      , numeroDivisores5 x]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_numeroDivisores\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> numeroDivisores1 (product [1..10])\n--    270\n--    (1.67 secs, 726,327,208 bytes)\n--    \u03bb> numeroDivisores2 (product [1..10])\n--    270\n--    (0.01 secs, 929,000 bytes)\n--\n--    \u03bb> numeroDivisores2 (product [1..16])\n--    5376\n--    (2.10 secs, 915,864,664 bytes)\n--    \u03bb> numeroDivisores3 (product [1..16])\n--    5376\n--    (0.01 secs, 548,472 bytes)\n--\n--    \u03bb> numeroDivisores3 (product [1..30])\n--    2332800\n--    (3.80 secs, 4,149,811,688 bytes)\n--    \u03bb> numeroDivisores4 (product [1..30])\n--    2332800\n--    (0.59 secs, 722,253,848 bytes)\n--    \u03bb> numeroDivisores5 (product [1..30])\n--    2332800\n--    (0.00 secs, 587,856 bytes)\n<\/pre>\n<p>El c\u00f3digo se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Exercitium\/blob\/main\/src\/Numero_de_divisores.hs\">GitHub<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Definir la funci\u00f3n numeroDivisores :: Integer -> Integer tal que (numeroDivisores x) es el n\u00famero de divisores de x. Por ejemplo, numeroDivisores 12 == 6 numeroDivisores 25 == 3 length (show (numeroDivisores (product [1..3*10^4]))) == 1948 Soluciones import Data.List (genericLength, group, inits) import Data.Numbers.Primes (primeFactors) import Test.QuickCheck &#8212; 1\u00aa soluci\u00f3n &#8212; =========== numeroDivisores1 :: Integer&#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":[41,8,12,498,501,282,368,258,13,74,10,89,11,247,157,6,482,236,521,146],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7018"}],"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=7018"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7018\/revisions"}],"predecessor-version":[{"id":7019,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7018\/revisions\/7019"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=7018"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=7018"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=7018"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}