{"id":4497,"date":"2019-01-03T06:00:35","date_gmt":"2019-01-03T04:00:35","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=4497"},"modified":"2019-01-10T09:23:46","modified_gmt":"2019-01-10T07:23:46","slug":"el-2019-es-semiprimo","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/el-2019-es-semiprimo\/","title":{"rendered":"El 2019 es semiprimo"},"content":{"rendered":"<p>Un <a href=\"http:\/\/bit.ly\/1NK8bJ0\">n\u00famero semiprimo<\/a> es un n\u00famero natural que es producto de dos n\u00fameros primos no necesariamente distintos. Por ejemplo, 26 es semiprimo (porque 26 = 2&#215;13) y 49 tambi\u00e9n lo es (porque 49 = 7&#215;7).<\/p>\n<p>Definir las funciones<\/p>\n<pre lang=\"text\">\n   esSemiprimo :: Integer -> Bool\n   semiprimos  :: [Integer]\n<\/pre>\n<p>tales que<\/p>\n<ul>\n<li>(esSemiprimo n) se verifica si n es semiprimo. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     esSemiprimo 26          ==  True\n     esSemiprimo 49          ==  True\n     esSemiprimo 8           ==  False\n     esSemiprimo 2019        ==  True\n     esSemiprimo (21+10^14)  ==  True\n<\/pre>\n<ul>\n<li>semiprimos es la sucesi\u00f3n de n\u00fameros semiprimos. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     take 10 semiprimos   ==  [4,6,9,10,14,15,21,22,25,26]\n     semiprimos !! 579    ==  2019\n     semiprimos !! 10000  ==  40886\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.Numbers.Primes \nimport Test.QuickCheck\n\n-- 1\u00aa definici\u00f3n de esSemiprimo\n-- ============================\n\nesSemiprimo :: Integer -> Bool\nesSemiprimo 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 definici\u00f3n de esSemiprimo\n-- ============================\n\nesSemiprimo2 :: Integer -> Bool\nesSemiprimo2 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 definici\u00f3n de esSemiprimo\n-- ============================\n\nesSemiprimo3 :: Integer -> Bool\nesSemiprimo3 n =\n  not (null [x | x <- reverse (takeWhile (<n) primes),\n                 n `mod` x == 0,\n                 isPrime (n `div` x)])\n\n-- 4\u00aa definici\u00f3n de esSemiprimo\n-- ============================\n\nesSemiprimo4 :: Integer -> Bool\nesSemiprimo4 n =\n  length (primeFactors n) == 2\n\n-- Equivalencia de las definiciones de esSemiprimo\n-- ===============================================\n\n-- La propiedad es\nprop_esSemiprimo :: Positive Integer -> Bool\nprop_esSemiprimo (Positive n) =\n  all (== esSemiprimo n) [f n | f <- [ esSemiprimo2\n                                     , esSemiprimo3\n                                     , esSemiprimo4\n                                     ]]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_esSemiprimo\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n--    \u03bb> esSemiprimo 5001\n--    True\n--    (1.90 secs, 274,450,648 bytes)\n--    \u03bb> esSemiprimo2 5001\n--    True\n--    (0.07 secs, 29,377,016 bytes)\n--    \u03bb> esSemiprimo3 5001\n--    True\n--    (0.01 secs, 1,706,840 bytes)\n--    \u03bb> esSemiprimo4 5001\n--    True\n--    (0.01 secs, 142,840 bytes)\n--    \n--    \u03bb> esSemiprimo2 100001\n--    True\n--    (2.74 secs, 1,473,519,064 bytes)\n--    \u03bb> esSemiprimo3 100001\n--    True\n--    (0.09 secs, 30,650,352 bytes)\n--    \u03bb> esSemiprimo4 100001\n--    True\n--    (0.01 secs, 155,200 bytes)\n--    \n--    \u03bb> esSemiprimo3 10000001\n--    True\n--    (8.73 secs, 4,357,875,016 bytes)\n--    \u03bb> esSemiprimo4 10000001\n--    True\n--    (0.01 secs, 456,328 bytes)\n\n-- Definici\u00f3n de semiprimos\n-- ========================\n\nsemiprimos :: [Integer]\nsemiprimos = filter esSemiprimo4 [4..]\n<\/pre>\n<h4>Pensamiento<\/h4>\n<blockquote><p>\nPorque toda visi\u00f3n requiere distancia, no hay manera de ver las cosas sin salirse de ellas. <\/p>\n<p>Antonio Machado\n<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>Un n\u00famero semiprimo es un n\u00famero natural que es producto de dos n\u00fameros primos no necesariamente distintos. Por ejemplo, 26 es semiprimo (porque 26 = 2&#215;13) y 49 tambi\u00e9n lo es (porque 49 = 7&#215;7). Definir las funciones esSemiprimo :: Integer -> Bool semiprimos :: [Integer] tales que (esSemiprimo n) se verifica si n 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":[5],"tags":[8,30,174,28,89,181,141,11,247,32,34],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4497"}],"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=4497"}],"version-history":[{"count":5,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4497\/revisions"}],"predecessor-version":[{"id":4531,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4497\/revisions\/4531"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=4497"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=4497"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=4497"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}