{"id":1265,"date":"2015-03-31T06:00:03","date_gmt":"2015-03-31T04:00:03","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=1265"},"modified":"2022-03-25T20:23:07","modified_gmt":"2022-03-25T18:23:07","slug":"pares-de-enteros-con-solo-un-factor-primo-comun","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/pares-de-enteros-con-solo-un-factor-primo-comun\/","title":{"rendered":"Pares de enteros con s\u00f3lo un factor primo com\u00fan"},"content":{"rendered":"<p>Dos enteros positivos a y b se dir\u00e1n relacionados si poseen, exactamente, un factor primo en com\u00fan. Por ejemplo, 12 y 20 est\u00e1n relacionados, pero 6 y 30 no lo est\u00e1n.<\/p>\n<p>Definir la lista infinita<\/p>\n<pre lang=\"text\">\n   paresRel :: [(Int,Int)]\n<\/pre>\n<p>tal que paresRel enumera todos los pares (a,b), con 1 \u2264 a &lt; b, tal que a y b est\u00e1n relacionados. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   ghci> take 10 paresRel\n   [(2,4),(2,6),(3,6),(4,6),(2,8),(4,8),(6,8),(3,9),(6,9),(2,10)]\n<\/pre>\n<p>\u00bfQu\u00e9 lugar ocupa el par (51,111) en la lista infinita paresRel?<\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (group, intersect, nub)\nimport Data.Numbers.Primes (primeFactors)\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nparesRel1 :: [(Int,Int)]\nparesRel1 = [(a,b) | b <- [1..], a <- [1..b-1], relacionados a b]\n\nrelacionados :: Int -> Int -> Bool\nrelacionados a b = \n    length (nub (primeFactors a `intersect` primeFactors b)) == 1\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nparesRel2 :: [(Int,Int)]\nparesRel2 = [(x,y) | y <- [4..], x <- [2..y-2], rel x y]\n    where rel x y = m \/= 1 &#038;&#038; all (== head ps) ps\n              where m  = gcd x y\n                    ps = primeFactors m\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\nparesRel3 :: [(Int,Int)]\nparesRel3 =\n    [(x,y) | y <- [2..], x <- [2..y-1], relacionados3 x y]\n\nrelacionados3 :: Int -> Int -> Bool\nrelacionados3 x y =\n    length (group (primeFactors (gcd x y))) == 1\n\n-- Comparaci\u00f3n de eficiencia\n--    \u03bb> paresRel1 !! 40000\n--    (216,489)\n--    (3.19 secs, 1,825,423,056 bytes)\n--    \u03bb> paresRel2 !! 40000\n--    (216,489)\n--    (0.96 secs, 287,174,864 bytes)\n--    \u03bb> paresRel3 !! 40000\n--    (216,489)\n--    (0.70 secs, 264,137,928 bytes)\n\n-- C\u00e1lculo\n-- =======\n\n-- El c\u00e1lculo es\n--    ghci> 1 + length (takeWhile (\/=(51,111)) paresRel1)\n--    2016\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Dos enteros positivos a y b se dir\u00e1n relacionados si poseen, exactamente, un factor primo en com\u00fan. Por ejemplo, 12 y 20 est\u00e1n relacionados, pero 6 y 30 no lo est\u00e1n. Definir la lista infinita paresRel :: [(Int,Int)] tal que paresRel enumera todos los pares (a,b), con 1 \u2264 a &lt; b, tal que a&#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":[5],"tags":[8,13,270,28,415,24,247],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/1265"}],"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=1265"}],"version-history":[{"count":6,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/1265\/revisions"}],"predecessor-version":[{"id":2051,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/1265\/revisions\/2051"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=1265"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=1265"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=1265"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}