{"id":2844,"date":"2017-01-23T06:00:58","date_gmt":"2017-01-23T04:00:58","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=2844"},"modified":"2017-01-30T09:41:01","modified_gmt":"2017-01-30T07:41:01","slug":"cadena-de-primos","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/cadena-de-primos\/","title":{"rendered":"Cadena de primos"},"content":{"rendered":"<p>La lista de los primeros n\u00fameros primos es<\/p>\n<pre lang=\"text\">\n   [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71]\n<\/pre>\n<p>Los primeros elementos de la cadena obtenida concatenado los n\u00fameros primos es<\/p>\n<pre lang=\"text\">\n   \"23571113171923293137414347535961677173798389971011\"\n<\/pre>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   primoEnPosicion :: Int -> Integer\n<\/pre>\n<p>tal que (primoEnPosicion n) es el n\u00famero primo que tiene alg\u00fan d\u00edgito en la posici\u00f3n n de la cadena obtenida concatenado los n\u00fameros primos. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   primoEnPosicion 0       ==  2\n   primoEnPosicion 1       ==  3\n   primoEnPosicion 4       ==  11\n   primoEnPosicion 5       ==  11\n   primoEnPosicion 6       ==  13\n   primoEnPosicion 1022    ==  2011\n   primoEnPosicion 1023    ==  2017\n   primoEnPosicion 1026    ==  2017\n   primoEnPosicion 1027    ==  2027\n   primoEnPosicion (10^7)  ==  21242357\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.Numbers.Primes\nimport Test.QuickCheck\n\n-- 1\u00aa definici\u00f3n\n-- =============\n\nprimoEnPosicion :: Int -> Integer\nprimoEnPosicion = aux primes\n  where aux (x:xs) n | m > n     = x\n                     | otherwise = aux xs (n-m)\n          where m = length (show x)\n\n\n-- 2\u00aa definici\u00f3n\n-- =============\n\nprimoEnPosicion2 :: Int -> Integer\nprimoEnPosicion2 n = p\n  where (p,_) = head $ dropWhile (\\(x,k) -> k < n) primosYfinales\n\n-- primosYfinales es la sucesi\u00f3n de los pares de los n\u00fameros primos y\n-- las posiciones de sus d\u00edgitos finales en la sucesi\u00f3n de la\n-- concatenaci\u00f3n de primos. Por ejemplo, \n--    \u03bb> take 10 primosYfinales\n--    [(2,0),(3,1),(5,2),(7,3),(11,5),(13,7),(17,9),(19,11),(23,13),(29,15)]\nprimosYfinales :: [(Integer, Int)]\nprimosYfinales = scanl f (2,0) (tail primes)\n  where f (p,k) q = (q, k + length (show q))\n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_equivalencia (Positive n) =\n  primoEnPosicion n == primoEnPosicion2 n\n\n-- La comprobaci\u00f3n es  \n--    \u03bb> quickCheck prop_equivalencia\n--    +++ OK, passed 100 tests.\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>La lista de los primeros n\u00fameros primos es [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71] Los primeros elementos de la cadena obtenida concatenado los n\u00fameros primos es \u00ab23571113171923293137414347535961677173798389971011\u00bb Definir la funci\u00f3n primoEnPosicion :: Int -> Integer tal que (primoEnPosicion n) es el n\u00famero primo que tiene alg\u00fan d\u00edgito en la posici\u00f3n n de la cadena obtenida concatenado los n\u00fameros primos. Por&#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":[59,71,28,11,173,6,78,33,45],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2844"}],"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=2844"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2844\/revisions"}],"predecessor-version":[{"id":2876,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2844\/revisions\/2876"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=2844"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=2844"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=2844"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}