{"id":2594,"date":"2016-11-18T06:00:59","date_gmt":"2016-11-18T04:00:59","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=2594"},"modified":"2016-11-25T06:56:16","modified_gmt":"2016-11-25T04:56:16","slug":"numeros-de-dudeney","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/numeros-de-dudeney\/","title":{"rendered":"N\u00fameros de Dudeney"},"content":{"rendered":"<p>La semana pasada, Pepe Mu\u00f1oz Santonja public\u00f3 en su blog <a href=\"http:\/\/algomasquenumeros.blogspot.com.es\">Algo m\u00e1s que n\u00fameros<\/a>  el art\u00edculo <a href=\"http:\/\/bit.ly\/2g1v8NQ\">N\u00fameros de Dudeney en la base OEIS<\/a><\/p>\n<p>Un <strong>n\u00famero de Dudeney<\/strong> es un n\u00famero entero n tal que el cubo de la suma de sus d\u00edgitos es igual a n. Por ejemplo, 512 es un n\u00famero de Dudeney ya que (5+1+2)^3 = 8^3 = 512.<\/p>\n<p>Se puede generalizar variando el exponente: Un <strong>n\u00famero de Dudeney de orden k<\/strong> es un n\u00famero entero n tal que la potencia k-\u00e9sima de la suma de sus d\u00edgitos es igual a n. Por ejemplo, 2401 es un n\u00famero de Dudeney de orden 4 ya que (2+4+0+1)^4 = 7^4 = 2401.<\/p>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   numerosDudeney :: Integer -> [Integer]\n<\/pre>\n<p>tal que (numerosDudeney k) es la lista de los n\u00fameros de Dudeney oe orden k. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   take 6 (numerosDudeney 3)  == [1,512,4913,5832,17576,19683]\n   take 6 (numerosDudeney 4)  == [1,2401,234256,390625,614656,1679616]\n   take 2 (numerosDudeney 20) == [1,1215766545905692880100000000000000000000]\n<\/pre>\n<p>Comprobar con QuickCheck que 19683 es el mayor n\u00famero de Dudeney de orden 3.<\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Test.QuickCheck\n\nnumerosDudeney :: Integer -> [Integer]\nnumerosDudeney k =\n  [n^k | n <- [1..]\n       , esDudeney k (n^k)]\n\nesDudeney :: Integer -> Integer -> Bool\nesDudeney k n =\n  (sum (digitos n)) ^ k == n\n\ndigitos :: Integer -> [Integer]\ndigitos n = [read [d] | d <- show n]\n\n-- La propiedad es\nprop_Dudeney :: Integer -> Property\nprop_Dudeney n =\n  n > 0 ==> not (esDudeney 3 (19683 + n))\n  \n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_Dudeney\n--    +++ OK, passed 100 tests.\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>La semana pasada, Pepe Mu\u00f1oz Santonja public\u00f3 en su blog Algo m\u00e1s que n\u00fameros el art\u00edculo N\u00fameros de Dudeney en la base OEIS Un n\u00famero de Dudeney es un n\u00famero entero n tal que el cubo de la suma de sus d\u00edgitos es igual a n. Por ejemplo, 512 es un n\u00famero de Dudeney ya&#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":[7],"tags":[8,95,33,40],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2594"}],"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=2594"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2594\/revisions"}],"predecessor-version":[{"id":2619,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2594\/revisions\/2619"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=2594"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=2594"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=2594"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}