{"id":3023,"date":"2017-02-28T06:00:47","date_gmt":"2017-02-28T04:00:47","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=3023"},"modified":"2017-03-07T07:39:08","modified_gmt":"2017-03-07T05:39:08","slug":"sucesiones-alicuotas","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/sucesiones-alicuotas\/","title":{"rendered":"Sucesiones al\u00edcuotas"},"content":{"rendered":"<p>La <a href=\"(http:\/\/bit.ly\/2lrJOam)\">sucesi\u00f3n al\u00edcuota<\/a> de un n\u00famero x es la sucesi\u00f3n cuyo primer t\u00e9rmino es x y cada otro t\u00e9rmino es la suma de los divisores propios del t\u00e9rmino anterior. Por ejemplo, la sucesi\u00f3n al\u00edcuota de 10 es [10,8,7,1,0,0,0] ya que<\/p>\n<pre lang=\"text\">\n   la suma de los divisores propios de 10 es 5 + 2 + 1 = 8\n   la suma de los divisores propios de  8 es 4 + 2 + 1 = 7\n   la suma de los divisores propios de  7 es 1\n   la suma de los divisores propios de  1 es 0\n<\/pre>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   sucAlicuota :: Integer -> [Integer]\n<\/pre>\n<p>tal que (sucAlicuota x) es la sucesi\u00f3n al\u00edcuota de x. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   take 6 (sucAlicuota 10)       ==  [10,8,7,1,0,0]\n   take 6 (sucAlicuota 95)       ==  [95,25,6,6,6,6]\n   take 6 (sucAlicuota 220)      ==  [220,284,220,284,220,284]\n   sucAlicuota 1184 !! (1+10^7)  ==  1210\n   sucAlicuota 276 !! 200        ==  2790456740340877466506\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Math.NumberTheory.ArithmeticFunctions (sigma)\n\n-- 1\u00aa definici\u00f3n\n-- =============\n\nsucAlicuota :: Integer -> [Integer]\nsucAlicuota x = x : sucAlicuota (sumaDivisoresPropios x)\n\nsumaDivisoresPropios :: Integer -> Integer\nsumaDivisoresPropios x =\n  sum [y | y <- [1..x-1], x `mod` y == 0]\n\n-- 2\u00aa definici\u00f3n\n-- =============\n\nsucAlicuota2 :: Integer -> [Integer]\nsucAlicuota2 = iterate (sum . divisoresPropios)\n\ndivisoresPropios :: Integer -> [Integer]\ndivisoresPropios x = [y | y <- [1..x-1], x `mod` y == 0]\n\n-- 3\u00aa definici\u00f3n\n-- =============\n\nsucAlicuota3 :: Integer -> [Integer]\nsucAlicuota3 x = aux x []\n  where aux y ys | y `elem` ys = us ++ cycle zs\n                 | otherwise   = aux (sumaDivisoresPropios y) (y:ys)\n          where us = reverse ys\n                zs = dropWhile (\/=y) us\n\n-- 4\u00aa definici\u00f3n\n-- =============\n\nsucAlicuota4 :: Integer -> [Integer]\nsucAlicuota4 x = x : sucAlicuota4 (sumaDivisoresPropios4 x)\n\nsumaDivisoresPropios4 :: Integer -> Integer\nsumaDivisoresPropios4 x =\n  sigma 1 x - x\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n--    \u03bb> sucAlicuota 1184 !! (1+10^3)\n--    1210\n--    (2.02 secs, 261,081,688 bytes)\n--    \u03bb> sucAlicuota2 1184 !! (1+10^3)\n--    1210\n--    (2.02 secs, 245,485,568 bytes)\n--    \u03bb> sucAlicuota3 1184 !! (1+10^3)\n--    1210\n--    (0.02 secs, 0 bytes)\n--    \u03bb> sucAlicuota4 1184 !! (1+10^3)\n--    1210\n--    (0.05 secs, 0 bytes)\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>La sucesi\u00f3n al\u00edcuota de un n\u00famero x es la sucesi\u00f3n cuyo primer t\u00e9rmino es x y cada otro t\u00e9rmino es la suma de los divisores propios del t\u00e9rmino anterior. Por ejemplo, la sucesi\u00f3n al\u00edcuota de 10 es [10,8,7,1,0,0,0] ya que la suma de los divisores propios de 10 es 5 + 2 + 1 =&#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":[8,166,59,26,89,11,6,32,379,40],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3023"}],"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=3023"}],"version-history":[{"count":5,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3023\/revisions"}],"predecessor-version":[{"id":3059,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3023\/revisions\/3059"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=3023"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=3023"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=3023"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}