{"id":4997,"date":"2019-05-13T06:00:48","date_gmt":"2019-05-13T04:00:48","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=4997"},"modified":"2021-04-25T12:40:15","modified_gmt":"2021-04-25T10:40:15","slug":"productos-simultaneos-de-dos-y-tres-numeros-consecutivos-2019","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/productos-simultaneos-de-dos-y-tres-numeros-consecutivos-2019\/","title":{"rendered":"Productos simult\u00e1neos de dos y tres n\u00fameros consecutivos"},"content":{"rendered":"<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   productos :: Integer -> Integer -> [[Integer]]\n<\/pre>\n<p>tal que (productos n x) es las listas de n elementos consecutivos cuyo producto es x. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   productos 2 6     ==  [[2,3]]\n   productos 3 6     ==  [[1,2,3]]\n   productos 4 1680  ==  [[5,6,7,8]]\n   productos 2 5     ==  []\n<\/pre>\n<p>Comprobar con QuickCheck que si n > 0 y x > 0, entonces<\/p>\n<pre lang=\"text\">\n   productos n (product [x..x+n-1]) == [[x..x+n-1]]\n<\/pre>\n<p>Usando productos, definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   productosDe2y3consecutivos :: [Integer]\n<\/pre>\n<p>cuyos elementos son los n\u00fameros naturales (no nulos) que pueden expresarse simult\u00e1neamente como producto de dos y tres n\u00fameros consecutivos. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   head productosDe2y3consecutivos  ==  6\n<\/pre>\n<p>Nota. Seg\u00fan demostr\u00f3 Mordell en 1962, productosDe2y3consecutivos s\u00f3lo tiene dos elementos.<\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Test.QuickCheck\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nproductos1 :: Integer -> Integer -> [[Integer]]\nproductos1 n x =\n  [[y..y+n-1] | y <- [1..x]\n              , product [y..y+n-1] == x]\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nproductos2 :: Integer -> Integer -> [[Integer]]\nproductos2 n x =\n  [[z..z+n-1] | z <- [1..y]\n              , product [z..z+n-1] == x]\n  where y = head (filter (\\y -> y^n >= x) [2..])\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> let (n,x) = (3,200) in productos1 n (product [x..x+n-1])\n--    [[200,201,202]]\n--    (7.38 secs, 7,235,956,440 bytes)\n--    \u03bb> let (n,x) = (3,200) in productos2 n (product [x..x+n-1])\n--    [[200,201,202]]\n--    (0.01 secs, 451,928 bytes)\n--\n--    \u03bb> productos2 3 1000018000107000210\n--    [[1000005,1000006,1000007]]\n--    (1.57 secs, 1,560,159,144 bytes)\n\n-- En lo que sigue se usa la 2\u00aa definici\u00f3n\nproductos :: Integer -> Integer -> [[Integer]]\nproductos = productos2\n\n-- La propiedad es\nprop_productos :: Integer -> Integer -> Property\nprop_productos n x =\n  n > 0 && x > 0 ==> productos n (product [x..x+n-1]) == [[x..x+n-1]]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_productos\n--    +++ OK, passed 100 tests.\n\nproductosDe2y3consecutivos :: [Integer]\nproductosDe2y3consecutivos =\n  [x | x <- [1..]\n     , let ys = productos 2 x\n     , not (null ys)\n     , let zs = productos 3 x\n     , not (null zs)] \n\n-- El c\u00e1lculo es\n--    \u03bb> take 2 productosDe2y3consecutivos\n--    [6,210]\n<\/pre>\n<h4>Pensamiento<\/h4>\n<blockquote><p>\nMis ojos en el espejo<br \/>\nson ojos ciegos que miran<br \/>\nlos ojos con que los veo.<\/p>\n<p>Antonio Machado\n<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>Definir la funci\u00f3n productos :: Integer -> Integer -> [[Integer]] tal que (productos n x) es las listas de n elementos consecutivos cuyo producto es x. Por ejemplo, productos 2 6 == [[2,3]] productos 3 6 == [[1,2,3]] productos 4 1680 == [[5,6,7,8]] productos 2 5 == [] Comprobar con QuickCheck que si n >&#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":[8,38,71,11,157],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4997"}],"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=4997"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4997\/revisions"}],"predecessor-version":[{"id":5034,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4997\/revisions\/5034"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=4997"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=4997"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=4997"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}