{"id":7118,"date":"2022-07-07T06:00:59","date_gmt":"2022-07-07T04:00:59","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=7118"},"modified":"2022-07-06T10:49:54","modified_gmt":"2022-07-06T08:49:54","slug":"mayor-semiprimo-menor-que-n","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/mayor-semiprimo-menor-que-n\/","title":{"rendered":"Mayor semiprimo menor que n"},"content":{"rendered":"<p>Un <a href=\"http:\/\/bit.ly\/1NK8bJ0\">n\u00famero semiprimo<\/a> es un n\u00famero natural  es producto de dos n\u00fameros primos no necesariamente distintos. Por ejemplo, 26 es semiprimo (porque 26 = 2\u00b713) y 49 tambi\u00e9n lo es (porque 49 = 7\u00b77).<\/p>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   mayorSemiprimoMenor :: Integer -> Integer\n<\/pre>\n<p>tal que <code>(mayorSemiprimoMenor n)<\/code> es el mayor semiprimo menor que <code>n<\/code> (suponiendo que <code>n<\/code> > 4). Por ejemplo,<\/p>\n<pre lang=\"text\">\n   mayorSemiprimoMenor 27      ==  26\n   mayorSemiprimoMenor 50      ==  49\n   mayorSemiprimoMenor 49      ==  46\n   mayorSemiprimoMenor (10^15) == 999999999999998\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.Numbers.Primes (primeFactors, isPrime, primes)\nimport Test.QuickCheck\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nmayorSemiprimoMenor1 :: Integer -> Integer\nmayorSemiprimoMenor1 n =\n  head [x | x <- [n-1,n-2..2], semiPrimo x]\n\nsemiPrimo :: Integer -> Bool\nsemiPrimo n =\n  not (null [x | x <- [n,n-1..2], \n                 primo x,\n                 n `mod` x == 0,\n                 primo (n `div` x)])\n\nprimo :: Integer -> Bool\nprimo n = [x | x <- [1..n], n `mod` x == 0] == [1,n] \n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nmayorSemiprimoMenor2 :: Integer -> Integer\nmayorSemiprimoMenor2 n =\n  head [x | x <- [n-1,n-2..2], semiPrimo2 x]\n\nsemiPrimo2 :: Integer -> Bool\nsemiPrimo2 n =\n  not (null [x | x <- [n-1,n-2..2], \n                 isPrime x,\n                 n `mod` x == 0,\n                 isPrime (n `div` x)])\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\nmayorSemiprimoMenor3 :: Integer -> Integer\nmayorSemiprimoMenor3 n =\n  head [x | x <- [n-1,n-2..2], semiPrimo3 x]\n\nsemiPrimo3 :: Integer -> Bool\nsemiPrimo3 n =\n  not (null [x | x <- reverse (takeWhile (<n) primes),\n                 n `mod` x == 0,\n                 isPrime (n `div` x)])\n\n-- 4\u00aa soluci\u00f3n\n-- ===========\n\nmayorSemiprimoMenor4 :: Integer -> Integer\nmayorSemiprimoMenor4 n =\n  head [ p | p <- [n-1,n-2..2]\n           , (length . primeFactors) p == 2]\n\n-- 5\u00aa soluci\u00f3n\n-- ===========\n\nmayorSemiprimoMenor5 :: Integer -> Integer\nmayorSemiprimoMenor5 n\n  | semiPrimo5 (n-1) = n-1\n  | otherwise        = mayorSemiprimoMenor5 (n-1)\n\nsemiPrimo5 :: Integer -> Bool\nsemiPrimo5 x = length (primeFactors x) == 2\n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_mayorSemiprimoMenor :: Integer -> Property\nprop_mayorSemiprimoMenor n =\n  n > 4 ==>\n  all (== mayorSemiprimoMenor1 n)\n      [mayorSemiprimoMenor2 n,\n       mayorSemiprimoMenor3 n,\n       mayorSemiprimoMenor4 n,\n       mayorSemiprimoMenor5 n]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_mayorSemiprimoMenor\n--    +++ OK, passed 100 tests; 353 discarded.\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> mayorSemiprimoMenor1 5000\n--    4997\n--    (1.92 secs, 945,507,880 bytes)\n--    \u03bb> mayorSemiprimoMenor2 5000\n--    4997\n--    (0.05 secs, 123,031,264 bytes)\n--    \u03bb> mayorSemiprimoMenor3 5000\n--    4997\n--    (0.01 secs, 5,865,120 bytes)\n--    \u03bb> mayorSemiprimoMenor4 5000\n--    4997\n--    (0.00 secs, 593,528 bytes)\n--    \u03bb> mayorSemiprimoMenor5 5000\n--    4997\n--    (0.00 secs, 593,200 bytes)\n--\n--    \u03bb> mayorSemiprimoMenor3 (3*10^6)\n--    2999995\n--    (2.34 secs, 6,713,620,000 bytes)\n--    \u03bb> mayorSemiprimoMenor4 (2*10^6)\n--    1999997\n--    (0.01 secs, 728,936 bytes)\n--    \u03bb> mayorSemiprimoMenor5 (2*10^6)\n--    1999997\n--    (0.01 secs, 728,608 bytes)\n<\/pre>\n<p>El c\u00f3digo se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Exercitium\/blob\/main\/src\/Mayor_semiprimo_menor_que_n.hs\">GitHub<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Un n\u00famero semiprimo es un n\u00famero natural es producto de dos n\u00fameros primos no necesariamente distintos. Por ejemplo, 26 es semiprimo (porque 26 = 2\u00b713) y 49 tambi\u00e9n lo es (porque 49 = 7\u00b77). Definir la funci\u00f3n mayorSemiprimoMenor :: Integer -> Integer tal que (mayorSemiprimoMenor n) es el mayor semiprimo menor que n (suponiendo que&#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":[2],"tags":[521],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7118"}],"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=7118"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7118\/revisions"}],"predecessor-version":[{"id":7119,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7118\/revisions\/7119"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=7118"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=7118"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=7118"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}