{"id":1155,"date":"2015-03-03T09:30:24","date_gmt":"2015-03-03T07:30:24","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=1155"},"modified":"2021-04-25T17:28:02","modified_gmt":"2021-04-25T15:28:02","slug":"productos-simultaneos-de-dos-y-tres-numeros-consecutivos-2015","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/productos-simultaneos-de-dos-y-tres-numeros-consecutivos-2015\/","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 definici\u00f3n\nproductos1 :: Integer -> Integer -> [[Integer]]\nproductos1 n x = [[y..y+n-1] | y <- [1..x],\n                               product [y..y+n-1] == x]\n\n-- 2\u00aa definici\u00f3n\n-- =============\n\n-- Se puede reducir el intervalo de b\u00fasqueda teniendo en cuenta las\n-- siguientes desigualdades\n--    y*(y+1)* ... (y+n-1) = x\n--    y^n <= x <= (y+n-1)^n\n--    y <= x^(1\/n), x^(1\/n)-n+1 <= y\n--    x^(1\/n)-n+1 <= y <= x^(1\/n)\n\nproductos2 :: Integer -> Integer -> [[Integer]]\nproductos2 n x = [[z..z+n-1] | z <- [y-n+1..y],\n                               product [z..z+n-1] == x]\n    where y = floor ((fromIntegral x)**(1\/(fromIntegral n)))\n\nproductos :: Integer -> Integer -> [[Integer]]\nproductos = productos2\n\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--    ghci> quickCheck prop_productos\n--    +++ OK, passed 100 tests.\n--    (0.10 secs, 26409644 bytes)\n\nproductosDe2y3consecutivos :: [Integer]\nproductosDe2y3consecutivos = [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--    ghci> take 2 productosDe2y3consecutivos\n--    [6,210]\n--    ghci> productos 2 210\n--    [[14,15]]\n--    ghci> productos 3 210\n--    [[5,6,7]]\n<\/pre>\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":[5],"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\/1155"}],"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=1155"}],"version-history":[{"count":5,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/1155\/revisions"}],"predecessor-version":[{"id":6358,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/1155\/revisions\/6358"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=1155"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=1155"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=1155"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}