{"id":4117,"date":"2018-05-29T06:00:17","date_gmt":"2018-05-29T04:00:17","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=4117"},"modified":"2018-06-05T16:32:49","modified_gmt":"2018-06-05T14:32:49","slug":"numeros-taxicab","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/numeros-taxicab\/","title":{"rendered":"N\u00fameros taxicab"},"content":{"rendered":"<p>Los <a href=\"http:\/\/bit.ly\/2ITYHhj\">n\u00fameros taxicab<\/a>, taxi-cab o n\u00fameros de Hardy-Ramanujan son aquellos n\u00fameros naturales que pueden expresarse como suma de dos cubos de m\u00e1s de una forma.<\/p>\n<p>Alternativamente, se define al n-\u00e9simo n\u00famero taxicab como el menor n\u00famero que es suma de dos cubos de n formas.<\/p>\n<p>Definir las siguientes sucesiones<\/p>\n<pre lang=\"text\">\n   taxicab  :: [Integer]\n   taxicab2 :: [Integer]\n<\/pre>\n<p>tales que taxicab es la sucesi\u00f3n de estos n\u00fameros seg\u00fan la primera definici\u00f3n y taxicab2 seg\u00fan la segunda. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   take 5 taxicab   ==  [1729,4104,13832,20683,32832]\n   take 2 taxicab2  ==  [2,1729]\n<\/pre>\n<p><strong>Nota 1<\/strong>. La sucesiones taxicab y taxicab2 se corresponden con las sucesiones <a href=\"https:\/\/oeis.org\/A001235\">A001235<\/a> y <a href=\"https:\/\/oeis.org\/A011541\">A011541<\/a> de la OEIS.<\/p>\n<p><strong>Nota 2<\/strong>: Este ejercicio ha sido propuesto por \u00c1ngel Ruiz Campos.<\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List  (find)\nimport Data.Maybe (fromJust)\n\ntaxicab :: [Integer]\ntaxicab = filter ((> 1) . length . descomposiciones) [1..]\n\n-- (descomposiciones n) es la lista de pares (x,y) tal que x\u00b3 + y\u00b3 = n.\n-- Por ejemplo,\n--    descomposiciones 1729  ==  [(10,9),(12,1)]\n--    descomposiciones 1728  ==  [(12,0)]\ndescomposiciones :: Integer -> [(Integer,Integer)]\ndescomposiciones n =\n  [(x,y) | x <- [1..round (fromIntegral n ** (1\/3))],\n           let z = n - x^3,\n           let z' = round (fromIntegral z ** (1\/3)),\n           z'^3 == z,\n           let y = z',\n           y <= x]\n\n-- 2\u00aa definici\u00f3n de descomposiciones\ndescomposiciones2 :: Integer -> [(Integer,Integer)]\ndescomposiciones2 n =\n  [(x,y) | x <- [1..raizEnt n 3],\n           let z = n - x^3,\n           let y = raizEnt z 3,\n           y <= x,    \n           y^3 == z]\n\n-- (raizEnt x n) es la ra\u00edz entera n-\u00e9sima de x; es decir, el mayor\n-- n\u00famero entero y tal que y^n <= x. Por ejemplo, \n--    raizEnt  8 3      ==  2\n--    raizEnt  9 3      ==  2\n--    raizEnt 26 3      ==  2\n--    raizEnt 27 3      ==  3\n--    raizEnt (10^50) 2 ==  10000000000000000000000000\n-- ---------------------------------------------------------------------\n\nraizEnt :: Integer -> Integer -> Integer\nraizEnt x n = aux (1,x)\n  where aux (a,b) | d == x    = c\n                  | c == a    = c\n                  | d < x     = aux (c,b)\n                  | otherwise = aux (a,c) \n          where c = (a+b) `div` 2\n                d = c^n\n\ntaxicab2 :: [Integer]\ntaxicab2 = aux 1 [2..]\n  where aux n xs = fx : aux (n+1) [fx+1..]\n          where fx = fromJust (find ((== n) . length . descomposiciones) xs)\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Los n\u00fameros taxicab, taxi-cab o n\u00fameros de Hardy-Ramanujan son aquellos n\u00fameros naturales que pueden expresarse como suma de dos cubos de m\u00e1s de una forma. Alternativamente, se define al n-\u00e9simo n\u00famero taxicab como el menor n\u00famero que es suma de dos cubos de n formas. Definir las siguientes sucesiones taxicab :: [Integer] taxicab2 :: [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":[4],"tags":[8,30,38,135,183,419,28,11,6,184],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4117"}],"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=4117"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4117\/revisions"}],"predecessor-version":[{"id":4135,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4117\/revisions\/4135"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=4117"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=4117"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=4117"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}