{"id":6026,"date":"2021-02-03T06:00:47","date_gmt":"2021-02-03T04:00:47","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=6026"},"modified":"2021-02-10T08:42:06","modified_gmt":"2021-02-10T06:42:06","slug":"los-numeros-armonicos-no-son-enteros","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/los-numeros-armonicos-no-son-enteros\/","title":{"rendered":"Los n\u00fameros arm\u00f3nicos no son enteros"},"content":{"rendered":"<p>Los <a href=\"https:\/\/bit.ly\/3a3PWQr\">n\u00fameros arm\u00f3nicos<\/a> son las sumas de los inversos de de los primeros n\u00fameros enteros positivos; es decir, el n-\u00e9simo n\u00famero arm\u00f3nico es<\/p>\n<pre lang=\"text\">\n   H(n) = 1 + 1\/2 + 1\/3 + \u00b7\u00b7\u00b7 + 1\/n\n<\/pre>\n<p>Los primeros n\u00fameros arm\u00f3nicos son<\/p>\n<pre lang=\"text\">\n   1, 3\/2, 11\/6, 25\/12, 137\/60, ..\n<\/pre>\n<p>Definir, usando la librer\u00eda de los n\u00fameros racionales (<a href=\"https:\/\/bit.ly\/3ogv1z1\">Data.Ratio<\/a>), las funciones<\/p>\n<pre lang=\"text\">\n   armonico  :: Integer -> Rational\n   armonicos :: [Rational]\n   esEntero  :: Rational -> Bool\n<\/pre>\n<p>tales que<\/p>\n<ul>\n<li>(armonico n) es el n-\u00e9simo n\u00famero arm\u00f3nico. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     armonico 2  ==    3 % 2\n     armonico 3  ==   11 % 6\n     armonico 4  ==   25 % 12\n     armonico 5  ==  137 % 60\n<\/pre>\n<ul>\n<li>armonicos es la lista de los n\u00fameros arm\u00f3nicos. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     take 5 armonicos  ==  [1 % 1,3 % 2,11 % 6,25 % 12,137 % 60]\n<\/pre>\n<ul>\n<li>(esEntero x) se verifica si x es un n\u00famero entero. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     esEntero (1 % 7)           ==  False\n     esEntero (1 % 7 + 20 % 7)  ==  True\n<\/pre>\n<p>Comprobar con QuickCheck que<\/p>\n<ul>\n<li>nig\u00fan n\u00famero arm\u00f3nico, excepto el primero, es un n\u00famero entero y<\/p>\n<\/li>\n<li>\n<p>la diferencia de dos n\u00fameros arm\u00f3nicos distintos nunca es un n\u00famero entero.<\/p>\n<\/li>\n<\/ul>\n<p><strong>Nota<\/strong>: Este ejercicio est\u00e1 basado en el art\u00edculo <a href=\"https:\/\/bit.ly\/3clIseL\">Sums of consecutive reciprocals<\/a> publicado por John D. Cook en su <a href=\"https:\/\/bit.ly\/3obB3kp\">blog<\/a> el 23 de enero de 2021.<\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (genericIndex)\nimport Data.Ratio ((%), denominator)\nimport Test.QuickCheck (Property, (==>), quickCheck)\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\narmonico  :: Integer -> Rational\narmonico 1 = 1\narmonico n = 1 % n + armonico (n-1)\n\narmonicos :: [Rational]\narmonicos = map armonico [1..]\n\nesEntero  :: Rational -> Bool\nesEntero x = denominator x == 1\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\narmonicos2 :: [Rational]\narmonicos2 = scanl1 (\\ x y -> x + y) [1 % n | n <- [1..]]\n\narmonico2  :: Integer -> Rational\narmonico2 n = armonicos `genericIndex` n\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> fromRational (armonicos !! (10^4))\n--    9.787706026045383\n--    (9.17 secs, 29,293,137,856 bytes)\n--    \u03bb> fromRational (armonicos2 !! (10^4))\n--    9.787706026045383\n--    (9.42 secs, 29,292,225,904 bytes)\n\n-- Propiedades\n-- ===========\n\n-- La 1\u00aa propiedad es\nprop_armonicos :: Integer -> Property\nprop_armonicos n =\n  n > 1 ==>\n  not (esEntero (armonico n))\n\n-- La comprobaci\u00f3n de la 1\u00aa propiedad es\n--    \u03bb> quickCheck prop_armonicos\n--    +++ OK, passed 100 tests.\n\n-- La 2\u00aa propiedad es\nprop_armonicos2 :: Integer -> Integer -> Property\nprop_armonicos2 n m =\n  n > 0 && m > 0 && n \/= m ==>\n  not (esEntero (armonico n - armonico m))\n\n-- La comprobaci\u00f3n de la segunda propiedad es\n--   \u03bb> quickCheck prop_armonicos2\n--   +++ OK, passed 100 tests.\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>Los n\u00fameros arm\u00f3nicos son las sumas de los inversos de de los primeros n\u00fameros enteros positivos; es decir, el n-\u00e9simo n\u00famero arm\u00f3nico es H(n) = 1 + 1\/2 + 1\/3 + \u00b7\u00b7\u00b7 + 1\/n Los primeros n\u00fameros arm\u00f3nicos son 1, 3\/2, 11\/6, 25\/12, 137\/60, .. Definir, usando la librer\u00eda de los n\u00fameros racionales (Data.Ratio), las&#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":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6026"}],"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=6026"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6026\/revisions"}],"predecessor-version":[{"id":6073,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6026\/revisions\/6073"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=6026"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=6026"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=6026"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}