{"id":5840,"date":"2021-02-24T11:42:45","date_gmt":"2021-02-24T10:42:45","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=5840"},"modified":"2021-04-28T11:06:41","modified_gmt":"2021-04-28T09:06:41","slug":"le-crible-deratosthene-en-python","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2021\/02\/24\/le-crible-deratosthene-en-python\/","title":{"rendered":"Le crible d\u2019\u00c9ratosth\u00e8ne en Python"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Le crible d\u2019\u00c9ratosth\u00e8ne en Python n&rsquo;est pas tr\u00e8s long \u00e0 impl\u00e9menter. Il existe sans doute plusieurs fa\u00e7ons de voir les choses; je vais vous en pr\u00e9senter une.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"278\" height=\"275\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2021\/02\/Eratosthene.01.png\" alt=\"crible \u00e9ratosth\u00e8ne Python\" class=\"wp-image-5842\"\/><figcaption>\u00c9ratosth\u00e8ne (image de Wikipedia)<\/figcaption><\/figure><\/div>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_89 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2021\/02\/24\/le-crible-deratosthene-en-python\/#Le_crible_dEratosthene_en_Python_la_logique\" >Le crible d\u2019\u00c9ratosth\u00e8ne en Python: la logique<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2021\/02\/24\/le-crible-deratosthene-en-python\/#Le_crible_dEratosthene_en_Python_le_programme\" >Le crible d\u2019\u00c9ratosth\u00e8ne en Python: le programme<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2021\/02\/24\/le-crible-deratosthene-en-python\/#Une_premiere_version\" >Une premi\u00e8re version<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2021\/02\/24\/le-crible-deratosthene-en-python\/#Une_deuxieme_version\" >Une deuxi\u00e8me version<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Le_crible_dEratosthene_en_Python_la_logique\"><\/span>Le crible d\u2019\u00c9ratosth\u00e8ne en Python: la logique<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Rappelons avant tout que le crible d\u2019\u00c9ratosth\u00e8ne consiste \u00e0 prendre la liste de tous les entiers de 2 \u00e0 un certain entier naturel <em>n<\/em>, puis \u00e0 supprimer tous les multiples des nombres rencontr\u00e9s.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On commence donc \u00e0 retenir \u00ab\u00a02\u00a0\u00bb et \u00e0 supprimer tous les nombres pairs (multiples de 2).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ensuite, on prend le premier entier qui suit \u00ab\u00a02\u00a0\u00bb et qui n&rsquo;a pas \u00e9t\u00e9 supprim\u00e9: c&rsquo;est \u00ab\u00a03\u00a0\u00bb. On supprime alors de la liste les multiples de 3 qui suivent.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">L&rsquo;entier suivant est \u00ab\u00a05\u00a0\u00bb; on le retient et on supprime tous les multiples de 5 qui restent. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Etc.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Le_crible_dEratosthene_en_Python_le_programme\"><\/span>Le crible d\u2019\u00c9ratosth\u00e8ne en Python: le programme<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">L&rsquo;id\u00e9e consiste donc \u00e0 prendre la liste de tous les entiers de 2 \u00e0 <em>n<\/em>, de la parcourir en supprimant tous les multiples des nombres rencontr\u00e9s. Facile non ? Ben&#8230; \u00e7a d\u00e9pend !<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Mon id\u00e9e est de construire une liste L = [ 2 , 3 , &#8230; , <em>n<\/em> ] et d&rsquo;initialiser une liste P vide pour y mettre tous les nombres rencontr\u00e9s qui ne seront pas barr\u00e9s.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Une_premiere_version\"><\/span>Une premi\u00e8re version<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">C&rsquo;est la premi\u00e8re version qui m&rsquo;est venue en t\u00eate:<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\"># Crible d\u2019\u00c9ratosth\u00e8ne - version 1\n\ndef eratosthene(n):\n    P = [ ]\n    for i in range(2,n+1):\n        if len(P) == 0:\n            P.append(i)\n        else:\n            prem = True\n            for k in P:\n                if i % k == 0:\n                    prem = False\n            if prem == True:\n                P.append(i)\n    \n    return P<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">qui donne par exemple:<\/p>\n\n\n\n<pre class=\"wp-block-preformatted\">&gt;&gt;&gt; eratosthene(100)\n[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]<\/pre>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Une_deuxieme_version\"><\/span>Une deuxi\u00e8me version<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u00c0 bien y r\u00e9fl\u00e9chir, la premi\u00e8re version n&rsquo;\u00e9tait pas exactement ce que je voulais faire. J&rsquo;ai donc \u00e9crit une autre version:<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\"># Crible d\u2019\u00c9ratosth\u00e8ne - version 2\n\ndef eratosthene(n):\n    L = [ i for i in range(2,n+1) ]\n    P = [ ]\n    \n    while len(L) != 0:\n        P.append(L[0])\n        i = L[0]\n        for k in L:\n            if k % i == 0:\n                del(L[L.index(k)])\n    \n    return P<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">La logique de cette version colle plus \u00e0 ce que je voulais faire, \u00e0 savoir parcourir la liste L et, pour chaque \u00e9l\u00e9ment rencontr\u00e9, le mettre dans la liste P et supprimer tous ses multiples. La logique est pr\u00e9cis\u00e9ment:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Tant que L est non vide, j&rsquo;ajoute \u00e0 P le premier \u00e9l\u00e9ment de L;<\/li><li>je parcours L et supprime tous les multiples de l&rsquo;\u00e9l\u00e9ment qui vient d&rsquo;\u00eatre ajout\u00e9 \u00e0 P.<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Et c&rsquo;est tout !<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">J&rsquo;ajouterai peut-\u00eatre d&rsquo;autres m\u00e9thodes  quand j&rsquo;aurai le temps&#8230; mais entretemps, vous pouvez aussi proposer votre version en commentaire (et je l&rsquo;ajouterai \u00e0 l&rsquo;article si elle ne ressemble pas d\u00e9j\u00e0 \u00e0 celles d\u00e9j\u00e0 propos\u00e9es).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Le crible d\u2019\u00c9ratosth\u00e8ne en Python n&rsquo;est pas tr\u00e8s long \u00e0 impl\u00e9menter. Il existe sans doute plusieurs fa\u00e7ons de voir les choses; je vais vous en pr\u00e9senter une.<\/p>\n","protected":false},"author":1,"featured_media":5842,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,5],"tags":[261,262],"class_list":["post-5840","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","category-python","tag-crible","tag-eratostene"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.8 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Le crible d\u2019\u00c9ratosth\u00e8ne en Python - Mathweb.fr - Plusieurs programmes<\/title>\n<meta name=\"description\" content=\"Le crible d\u2019\u00c9ratosth\u00e8ne en Python n&#039;est pas tr\u00e8s long \u00e0 impl\u00e9menter, mais il force notre fa\u00e7on de voir les choses \u00e0 \u00eatre claire. 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