{"id":6320,"date":"2021-04-28T06:00:41","date_gmt":"2021-04-28T04:00:41","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=6320"},"modified":"2021-05-05T08:05:21","modified_gmt":"2021-05-05T06:05:21","slug":"numeros-iguales-a-potencias-de-las-sumas-de-sus-cifras-ome1999-p2","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/numeros-iguales-a-potencias-de-las-sumas-de-sus-cifras-ome1999-p2\/","title":{"rendered":"N\u00fameros iguales a potencias de las sumas de sus cifras (OME1999 P2)"},"content":{"rendered":"<p>El enunciado del <a href=\"https:\/\/bit.ly\/3aqnPMq\">problema 2 de la OME (Olimpiada Matem\u00e1tica Espa\u00f1ola) del 1998<\/a> es<\/p>\n<blockquote><p>\n  Hallar todos los n\u00fameros naturales de 4 cifras, escritos en base 10, que sean iguales al cubo de la suma de sus cifras.\n<\/p><\/blockquote>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   especiales :: Integer -> Integer -> [Integer]\n<\/pre>\n<p>tal que (especiales a b) es la lista de los n\u00fameros de a cifras que son iguales la suma de sus cifras elevada a b. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   especiales 5 3  ==  [17576,19683]\n   especiales 6 4  ==  [234256,390625,614656]\n<\/pre>\n<p>Usando la funci\u00f3n anterior, calcular las soluciones del problema de la Olimpiada.<\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.Char (digitToInt)\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nespeciales :: Integer -> Integer -> [Integer]\nespeciales a b =\n  [n | n <- [10^(a-1)..10^a-1],\n       n == (sumaDigitos n)^b]\n\n-- (sumaDigitos n) es la suma de los d\u00edgitos de n. Por ejemplo,\n--    sumaDigitos 2021  ==  5\nsumaDigitos :: Integer -> Integer\nsumaDigitos = sum . digitos\n\n-- (digitos n) es la lista de los d\u00edgitos de n. Por ejemplo,\n--    digitos 2021  ==  [2,0,2,1]\ndigitos :: Integer -> [Integer]\ndigitos x = [read [c] | c <- show x]\n\n-- C\u00e1lculo de la soluci\u00f3n del problema de la Olimpiada:\n--    \u03bb> especiales 4 3\n--    [4913,5832]\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nespeciales2 :: Integer -> Integer -> [Integer]\nespeciales2 c e = [z | z <- map (^e) [a..min (9*c) b], z == (f z)^e]\n  where [c', e'] = map fromIntegral [c,e]\n        [a , b ] = [ceiling (10**((c'+p)\/e')) | p <- [-1,0]]\n        f = sum . map (toInteger . digitToInt) . show\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> especiales 6 4\n--    [234256,390625,614656]\n--    (7.41 secs, 21,808,919,208 bytes)\n--    \u03bb> especiales2 6 4\n--    [234256,390625,614656]\n--    (0.01 secs, 168,072 bytes)\n<\/pre>\n<h4>Nuevas soluciones<\/h4>\n<ul>\n<li>En los comentarios se pueden escribir nuevas soluciones.\n<li>El c\u00f3digo se debe escribir entre una l\u00ednea con &#60;pre lang=&quot;haskell&quot;&#62; y otra con &#60;\/pre&#62;\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>El enunciado del problema 2 de la OME (Olimpiada Matem\u00e1tica Espa\u00f1ola) del 1998 es Hallar todos los n\u00fameros naturales de 4 cifras, escritos en base 10, que sean iguales al cubo de la suma de sus cifras. Definir la funci\u00f3n especiales :: Integer -> Integer -> [Integer] tal que (especiales a b) es la lista&#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":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6320"}],"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=6320"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6320\/revisions"}],"predecessor-version":[{"id":6401,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6320\/revisions\/6401"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=6320"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=6320"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=6320"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}