{"id":2166,"date":"2016-02-25T06:00:50","date_gmt":"2016-02-25T04:00:50","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=2166"},"modified":"2016-03-03T06:46:09","modified_gmt":"2016-03-03T04:46:09","slug":"numeros-automorficos","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/numeros-automorficos\/","title":{"rendered":"N\u00fameros autom\u00f3rficos"},"content":{"rendered":"<p>Un n\u00famero n es <strong>autom\u00f3rfico<\/strong> si los \u00faltimos d\u00edgitos de su cuadrado son los d\u00edgitos de n. Por ejemplo, 5, 6, 76 y 890625 son n\u00fameros autom\u00f3rficos ya que 5\u00b2 = 2<strong>5<\/strong>, 6\u00b2 = 3<strong>6<\/strong>, 76\u00b2 = 57<strong>76<\/strong> y  890625\u00b2 = 7932128<strong>90625<\/strong>.<\/p>\n<p>Definir la sucesi\u00f3n<\/p>\n<pre lang=\"text\">\n   automorficos :: [Integer]\n<\/pre>\n<p>tal que sus elementos son los n\u00fameros autom\u00f3rficos. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   \u03bb> take 11 automorficos\n   [1,5,6,25,76,376,625,9376,90625,109376,890625]\n   \u03bb> automorficos !! 30\n   56259918212890625\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (isSuffixOf, nub, sort)\n\nautomorficos :: [Integer] \nautomorficos = filter esAutomorfico [1..]\n\nesAutomorfico :: Integer -> Bool \nesAutomorfico n = show n `isSuffixOf` show (n*n)\n\n-- 2\u00aa definici\u00f3n \n-- =============\n\nautomorficos2 :: [Integer] \nautomorficos2 = nub (1 : concat [sort [a,b] |\n                                 k <- [1..], \n                                 let a = 5^(2^k) `mod` 10^k, \n                                 let b = 10^k - a + 1])\n\n-- Comparaci\u00f3n de eficiencia \n-- =========================\n\n-- \u03bb> automorficos !! 12 \n-- 7109376 \n-- (16.64 secs, 6,759,638,824 bytes)\n-- \u03bb> automorficos2 !! 12 \n-- 7109376 \n-- (0.00 secs, 0 bytes)\n<\/pre>\n<h4>Referencias<\/h4>\n<ul>\n<li>J.D. Cook, <a href=\"http:\/\/bit.ly\/1mYTYiu\">Curious numbers<\/a>.<\/li>\n<li>Wikipedia, <a href=\"http:\/\/bit.ly\/1mYTZCW\">Automorphic number<\/a>.<\/li>\n<li>E.W. Weisstein, <a href=\"\">Automorphic number<\/a> en MathWorld.<\/li>\n<li>N.J.A. Sloane <a href=\"https:\/\/oeis.org\/A003226\">Sucesion A003226<\/a> en OEIS.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Un n\u00famero n es autom\u00f3rfico si los \u00faltimos d\u00edgitos de su cuadrado son los d\u00edgitos de n. Por ejemplo, 5, 6, 76 y 890625 son n\u00fameros autom\u00f3rficos ya que 5\u00b2 = 25, 6\u00b2 = 36, 76\u00b2 = 5776 y 890625\u00b2 = 793212890625. Definir la sucesi\u00f3n automorficos :: [Integer] tal que sus elementos son los n\u00fameros&#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\/2166"}],"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=2166"}],"version-history":[{"count":5,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2166\/revisions"}],"predecessor-version":[{"id":2197,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2166\/revisions\/2197"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=2166"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=2166"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=2166"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}