{"id":3410,"date":"2017-11-15T06:00:02","date_gmt":"2017-11-15T04:00:02","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=3410"},"modified":"2017-11-25T13:34:04","modified_gmt":"2017-11-25T11:34:04","slug":"numeros-digito-potenciales","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/numeros-digito-potenciales\/","title":{"rendered":"N\u00fameros d\u00edgito potenciales"},"content":{"rendered":"<p>Un n\u00famero entero x es <em>d\u00edgito potencial<\/em> de orden n si x es la suma de los d\u00edgitos de x elevados a n. Por ejemplo,<\/p>\n<ul>\n<li>153 es un d\u00edgito potencial de orden 3 ya que 153 = 1^3+5^3+3^3<\/li>\n<li>4150 es un d\u00edgito potencial de orden 5 ya que 4150 = 4^5+1^5+5^5+0^5<\/li>\n<\/ul>\n<p>Un n\u00famero x es <em>d\u00edgito auto potencial<\/em> si es un d\u00edgito potencial de orden n, donde n es el n\u00famero de d\u00edgitos de n. Por ejemplo, 153 es un n\u00famero d\u00edgito auto potencial.<\/p>\n<p>Definir las funciones<\/p>\n<pre lang=\"text\">\n   digitosPotencialesOrden :: Integer -> [Integer]\n   digitosAutoPotenciales  :: [Integer]\n<\/pre>\n<p>tales que<\/p>\n<ul>\n<li>(digitosPotencialesOrden n) es la lista de los n\u00fameros d\u00edgito potenciales de orden n. Por ejemplo, <\/li>\n<\/ul>\n<pre lang=\"text\">\n     take 6 (digitosPotencialesOrden 3)  ==  [0,1,153,370,371,407]\n     take 5 (digitosPotencialesOrden 4)  ==  [0,1,1634,8208,9474]\n     take 8 (digitosPotencialesOrden 5)  ==  [0,1,4150,4151,54748,92727,93084,194979]\n     take 3 (digitosPotencialesOrden 6)  ==  [0,1,548834]\n<\/pre>\n<ul>\n<li>digitosAutoPotenciales es la lista de los n\u00fameros d\u00edgito auto potenciales. Por ejemplo, <\/li>\n<\/ul>\n<pre lang=\"text\">\n     \u03bb> take 20 digitosAutoPotenciales\n     [0,1,2,3,4,5,6,7,8,9,153,370,371,407,1634,8208,9474,54748,92727,93084]\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (genericLength)\nimport Data.Char (digitToInt)\n\n-- 1\u00aa definici\u00f3n de digitosPotencialesOrden\n-- ========================================\n\ndigitosPotencialesOrden :: Integer -> [Integer]\ndigitosPotencialesOrden n =\n  [x | x <- [0..]\n     , esDigitoPotencialOrden n x]\n\nesDigitoPotencialOrden :: Integer -> Integer -> Bool\nesDigitoPotencialOrden n x =\n  x == sum [y^n | y <- digitos x]\n\ndigitos :: Integer -> [Integer]\ndigitos x = [read [d] | d <- show x] \n\n-- 2\u00aa definici\u00f3n de digitosPotencialesOrden\n-- ========================================\n\ndigitosPotencialesOrden2 :: Integer -> [Integer]\ndigitosPotencialesOrden2 n =\n  filter (esDigitoPotencialOrden2 n) [0..]\n\nesDigitoPotencialOrden2 :: Integer -> Integer -> Bool\nesDigitoPotencialOrden2 n x =\n  x == sum (map (^n) (digitos2 x))\n\ndigitos2 :: Integer -> [Integer]\ndigitos2 = map (toInteger . digitToInt) . show\n\n-- 3\u00aa definici\u00f3n de digitosPotencialesOrden\n-- ========================================\n\n--    digitosPotencialesOrden3 3  ==  [0,1,153,370,371,407]\n--    digitosPotencialesOrden3 4  ==  [0,1,1634,8208,9474]\ndigitosPotencialesOrden3 :: Integer -> [Integer]\ndigitosPotencialesOrden3 n =\n  filter (esDigitoPotencialOrden2 n) [0..10^d-1]\n  where d = maximoNDigitosPotencialesOrden n \n\n-- (maximoNDigitosPotencialesOrden n) es el m\u00e1ximo n\u00famero de d\u00edgitos de\n-- los n\u00fameros d\u00edgitos potenciales de orden d. Por ejemplo,\n--    maximoNDigitosPotencialesOrden 3  ==  5\n--    maximoNDigitosPotencialesOrden 5  ==  7\nmaximoNDigitosPotencialesOrden :: Integer -> Integer\nmaximoNDigitosPotencialesOrden n =\n  head (dropWhile (\\d -> d*9^n >= 10^(d-1)) [1..])\n\n-- 1\u00aa definici\u00f3n de esDigitoAutoPotencial\n-- ======================================\n\nesDigitoAutoPotencial :: Integer -> Bool\nesDigitoAutoPotencial x =\n  esDigitoPotencialOrden (genericLength (show x)) x\n\ndigitosAutoPotenciales :: [Integer]\ndigitosAutoPotenciales =\n  filter esDigitoAutoPotencial [0..]\n\n-- 2\u00aa definici\u00f3n de esDigitoAutoPotencial\n-- ======================================\n\ndigitosAutoPotenciales2 :: [Integer]\ndigitosAutoPotenciales2 =\n  0: concat [[x | x <- [10^k..10^(k+1)-1], esDigitoPotencialOrden (k+1) x]\n            | k <- [0..]]\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Un n\u00famero entero x es d\u00edgito potencial de orden n si x es la suma de los d\u00edgitos de x elevados a n. Por ejemplo, 153 es un d\u00edgito potencial de orden 3 ya que 153 = 1^3+5^3+3^3 4150 es un d\u00edgito potencial de orden 5 ya que 4150 = 4^5+1^5+5^5+0^5 Un n\u00famero x es&#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":[8,12,248,59,38,258,71,10,11,95,33,410],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3410"}],"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=3410"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3410\/revisions"}],"predecessor-version":[{"id":3446,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3410\/revisions\/3446"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=3410"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=3410"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=3410"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}