{"id":4437,"date":"2018-12-19T06:00:53","date_gmt":"2018-12-19T04:00:53","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=4437"},"modified":"2018-12-26T09:40:01","modified_gmt":"2018-12-26T07:40:01","slug":"grado-exponencial","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/grado-exponencial\/","title":{"rendered":"Grado exponencial"},"content":{"rendered":"<p>El grado exponencial de un n\u00famero n es el menor n\u00famero x mayor que 1 tal que n es una subcadena de <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=n%5Ex&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"n^x\" class=\"latex\" \/>. Por ejemplo, el grado exponencial de 2 es 5 ya que 2 es una subcadena de 32 (que es <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=2%5E5&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"2^5\" class=\"latex\" \/>) y no es subcadena de las anteriores potencias de 2 (2, 4 y 16). El grado exponencial de 25 es 2 porque 25 es una subcadena de 625 (que es <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=25%5E2&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"25^2\" class=\"latex\" \/>).<\/p>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   gradoExponencial :: Integer -> Integer\n<\/pre>\n<p>tal que (gradoExponencial n) es el grado exponencial de n. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   gradoExponencial 2      ==  5\n   gradoExponencial 25     ==  2\n   gradoExponencial 15     ==  26\n   gradoExponencial 1093   ==  100\n   gradoExponencial 10422  ==  200\n   gradoExponencial 11092  ==  300\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Test.QuickCheck\nimport Data.List (genericLength, isInfixOf)\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\ngradoExponencial :: Integer -> Integer\ngradoExponencial n =\n  head [e | e <- [2..]\n          , show n `isInfixOf` show (n^e)]\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\ngradoExponencial2 :: Integer -> Integer\ngradoExponencial2 n =\n  2 + genericLength (takeWhile noSubcadena (potencias n))\n  where c             = show n\n        noSubcadena x = not (c `isInfixOf`show x)\n\n-- (potencias n) es la lista de las potencias de n a partir de n^2. Por\n-- ejemplo, \n--    \u03bb> take 10 (potencias 2)\n--    [4,8,16,32,64,128,256,512,1024,2048]\npotencias :: Integer -> [Integer]\npotencias n =\n  iterate (*n) (n^2)\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\ngradoExponencial3 :: Integer -> Integer\ngradoExponencial3 n = aux 2\n  where aux x\n          | cs `isInfixOf` show (n^x) = x\n          | otherwise                 = aux (x+1)\n        cs = show n\n\n-- Equivalencia\n-- ============\n\n-- La propiedad es\nprop_gradosExponencial_equiv :: (Positive Integer) -> Bool\nprop_gradosExponencial_equiv (Positive n) =\n  gradoExponencial n == gradoExponencial2 n &&\n  gradoExponencial n == gradoExponencial3 n\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_gradosExponencial_equiv\n--    +++ OK, passed 100 tests.\n<\/pre>\n<h4>Referencia<\/h4>\n<p>Basado en la <a href=\"https:\/\/oeis.org\/A045537\">sucesi\u00f3n A045537 de la OEIS<\/a>.<\/p>\n<h4>Pensamiento<\/h4>\n<blockquote><p>\n\u00abDe cada diez novedades que pretenden descubrirnos, nueve son<br \/>\ntonter\u00edas. La d\u00e9cima y \u00faltima, que no es necedad, resulta a \u00faltima hora<br \/>\nque tampoco es nueva.\u00bb<\/p>\n<p>Antonio Machado\n<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>El grado exponencial de un n\u00famero n es el menor n\u00famero x mayor que 1 tal que n es una subcadena de . Por ejemplo, el grado exponencial de 2 es 5 ya que 2 es una subcadena de 32 (que es ) y no es subcadena de las anteriores potencias de 2 (2, 4&#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":[8,258,316,50,181,11,33,34],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4437"}],"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=4437"}],"version-history":[{"count":6,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4437\/revisions"}],"predecessor-version":[{"id":4480,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4437\/revisions\/4480"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=4437"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=4437"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=4437"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}