{"id":2976,"date":"2020-07-23T16:34:20","date_gmt":"2020-07-23T14:34:20","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=2976"},"modified":"2020-10-03T15:20:54","modified_gmt":"2020-10-03T13:20:54","slug":"particularites-numeriques-constantes-kaprekar","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/","title":{"rendered":"Particularit\u00e9s num\u00e9riques (constantes de Kaprekar)"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Nous allons parler dans cet article, inspir\u00e9 de la vid\u00e9o suivante, des constantes de Kaprekar et aller un peu plus loin.<\/p>\n\n\n\n<figure class=\"wp-block-embed-youtube wp-block-embed is-type-video is-provider-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"MAGICAL Indian Math Discovery - Numbers 495 and 6174 (Kaprekar Constants)\" width=\"750\" height=\"422\" src=\"https:\/\/www.youtube.com\/embed\/qX9RIU-FrYk?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<!--more-->\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"392\" height=\"512\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/07\/kaprekar.jpg\" alt=\"constantes de Kaprekar\" class=\"wp-image-2979\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/07\/kaprekar.jpg 392w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/07\/kaprekar-300x392.jpg 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/07\/kaprekar-230x300.jpg 230w\" sizes=\"auto, (max-width: 392px) 100vw, 392px\" \/><figcaption>Kaprekar<\/figcaption><\/figure><\/div>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_87 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\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/#Le_probleme_mathematique_sur_un_nombre_a_3_chiffres_pour_obtenir_une_des_constantes_de_Kaprekar_495\" >Le probl\u00e8me math\u00e9matique sur un nombre \u00e0 3 chiffres pour obtenir une des constantes de Kaprekar 495<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/#Expose_theorique\" >Expos\u00e9 th\u00e9orique<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/#Un_exemple\" >Un exemple<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/#Le_probleme_mathematique_sur_un_nombre_a_4_chiffres_pour_obtenir_une_des_constantes_de_Kaprekar_6174\" >Le probl\u00e8me math\u00e9matique sur un nombre \u00e0 4 chiffres pour obtenir une des constantes de Kaprekar 6174<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/#Un_programme_Python\" >Un programme Python<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/#Le_probleme_mathematique_sur_un_nombre_a_5_chiffres_et_plus_constantes_de_Kaprekar\" >Le probl\u00e8me math\u00e9matique sur un nombre \u00e0 5 chiffres et plus : constantes de Kaprekar ?<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/#Constat_mathematique\" >Constat math\u00e9matique<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/#Modification_du_programme_Python\" >Modification du programme Python<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Le_probleme_mathematique_sur_un_nombre_a_3_chiffres_pour_obtenir_une_des_constantes_de_Kaprekar_495\"><\/span>Le probl\u00e8me math\u00e9matique sur un nombre \u00e0 3 chiffres pour obtenir une des constantes de Kaprekar 495<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Expose_theorique\"><\/span>Expos\u00e9 th\u00e9orique<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u00c9tant donn\u00e9 un nombre entier \\(n\\) \u00e0 3 chiffres non tous identiques, not\u00e9 \\(\\overline{x_1x_2x_3}\\), on consid\u00e8re les fonctions:$$b(n)=\\overline{x_ix_jx_k},~\\text{avec }x_i \\geqslant x_j \\geqslant x_k$$et:$$s(n)=\\overline{x_mx_nx_p},~\\text{avec }x_m \\leqslant x_n \\leqslant x_p.$$On note ensuite:$$d(n)=b(n)-s(n).$$Alors, il existe un entier <em>r<\/em>&gt;0 tel que pour tout entier \\(q \\geqslant r\\), \\(d^q(n)=495\\), en prenant pour convention que:$$d^q(n) = \\underbrace{(d \\circ d \\circ \\cdots \\circ d)}_{q\\text{ fois}}(n).$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Un_exemple\"><\/span>Un exemple<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Prenons:$$n=847$$. Alors,$$b(n)=874$$et$$s(n)=478$$. Ici, on forme le plus grand entier possible avec les chiffres de <em>n<\/em> (c&rsquo;est <em>b<\/em>(<em>n<\/em>) &#8211; <em>b<\/em> pour \u00ab\u00a0biggest\u00a0\u00bb) et le plus petit nombre possible avec ces m\u00eames chiffres (c&rsquo;est <em>s<\/em>(<em>n<\/em>) &#8211; <em>s<\/em> pour \u00ab\u00a0smallest\u00a0\u00bb).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ensuite, on calcule la diff\u00e9rence:$$d(n)=874-478=396.$$On applique alors \u00e0 396 la fonction <em>d<\/em> \u00e0 nouveau. On trouve:$$d(396)=963-369=594,$$ puis on applique \u00e0 nouveau <em>d<\/em> \u00e0 594:$$d(594)=954-459=495.$$Si on applique \u00e0 nouveau <em>d<\/em> \u00e0 495, on trouve:$$d(495)=954-459=495.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ainsi, pour tout entier \\(q \\geqslant 3\\),$$d^q(847)=495.$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Le_probleme_mathematique_sur_un_nombre_a_4_chiffres_pour_obtenir_une_des_constantes_de_Kaprekar_6174\"><\/span>Le probl\u00e8me math\u00e9matique sur un nombre \u00e0 4 chiffres pour obtenir une des constantes de Kaprekar 6174<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Si on fait la m\u00eame chose sur un nombre \u00e0 quatre chiffres, on constate qu&rsquo;il existe un entier <em>r<\/em> \u00e0 partir duquel \\(d^q(n)=6174\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Par exemple, si on prend:$$n=4521$$on a:$$d(n)=5421-1245=4176$$et$$d^2(n)=7641-1467=6174.$$On peut voir aussi que \\(d(6174)=6174\\) donc inutile d&rsquo;aller plus loin.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Un_programme_Python\"><\/span>Un programme Python<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Comme je suis une grosse feignasse, je  n&rsquo;aime pas faire de calculs r\u00e9p\u00e9titifs donc je pr\u00e9f\u00e8re \u00e9crire un programme qui fait \u00e7a pour moi&#8230;<\/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=\"\">def d(n):\n    L = list( str(n) )\n    L.sort()\n    smallest = ''.join(L)\n    L.reverse()\n    biggest = ''.join( L )\n    return int(smallest) , int(biggest) , int(biggest)-int(smallest)\n\nn = int( input(\"Entrez un nombre entier : \") )\nant = 0\n\nwhile d(ant) != d(n):\n    print('{} --> {} , {} --> diff\u00e9rence = {}'.format(n , d(n)[0] , d(n)[1] , d(n)[2]))\n    ant = n\n    n = d(ant)[2]<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Ce n&rsquo;est sans doute pas un programme optimal, mais&#8230; apr\u00e8s tout, je suis en vacances ! Et il fait le job&#8230;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">L&rsquo;id\u00e9e est de voir les diff\u00e9rents r\u00e9sultats de \\(d^q(n)\\), et \u00e7a fonctionne bien. Bien s\u00fbr, en l&rsquo;\u00e9tat, si vous entrez un entier \u00e0 plus de 4 chiffres, \u00e7a ne fa pas fonctionner&#8230;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Le_probleme_mathematique_sur_un_nombre_a_5_chiffres_et_plus_constantes_de_Kaprekar\"><\/span>Le probl\u00e8me math\u00e9matique sur un nombre \u00e0 5 chiffres et plus : constantes de Kaprekar ?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Constat_mathematique\"><\/span>Constat math\u00e9matique<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Si on prend un nombre \u00e0 5 chiffres, on s&rsquo;aper\u00e7oit que les choses changent. \u00c0 l&rsquo;aide du programme pr\u00e9c\u00e9dent, on le stoppant manuellement, on s\u2019aper\u00e7oit qu&rsquo;il y a un cycle.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Par exemple, si on prend:$$n=12345$$cela donne:<\/p>\n\n\n\n<pre class=\"wp-block-preformatted\">12345 --&gt; 12345 , 54321 --&gt; diff\u00e9rence = 41976\n41976 --&gt; 14679 , 97641 --&gt; diff\u00e9rence = 82962 &lt;====\n82962 --&gt; 22689 , 98622 --&gt; diff\u00e9rence = 75933\n75933 --&gt; 33579 , 97533 --&gt; diff\u00e9rence = 63954\n63954 --&gt; 34569 , 96543 --&gt; diff\u00e9rence = 61974\n61974 --&gt; 14679 , 97641 --&gt; diff\u00e9rence = 82962 &lt;==== cycle<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">En prenant:$$n=28018$$on a:<\/p>\n\n\n\n<pre class=\"wp-block-preformatted\">28018 --&gt; 1288 , 88210 --&gt; diff\u00e9rence = 86922\n86922 --&gt; 22689 , 98622 --&gt; diff\u00e9rence = 75933 &lt;===\n75933 --&gt; 33579 , 97533 --&gt; diff\u00e9rence = 63954\n63954 --&gt; 34569 , 96543 --&gt; diff\u00e9rence = 61974\n61974 --&gt; 14679 , 97641 --&gt; diff\u00e9rence = 82962\n82962 --&gt; 22689 , 98622 --&gt; diff\u00e9rence = 75933 &lt;=== cycle<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">En prenant:$$n=98174$$on a:<\/p>\n\n\n\n<pre class=\"wp-block-preformatted\">98174 --&gt; 14789 , 98741 --&gt; diff\u00e9rence = 83952 &lt;===\n83952 --&gt; 23589 , 98532 --&gt; diff\u00e9rence = 74943\n74943 --&gt; 34479 , 97443 --&gt; diff\u00e9rence = 62964\n62964 --&gt; 24669 , 96642 --&gt; diff\u00e9rence = 71973\n71973 --&gt; 13779 , 97731 --&gt; diff\u00e9rence = 83952 &lt;=== cycle<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Bon, on ne va pas tous les faire mais bon&#8230; allez ! Un dernier et j&rsquo;arr\u00eate: pour$$n=10101$$on a:<\/p>\n\n\n\n<pre class=\"wp-block-preformatted\">10101 --&gt; 111 , 11100 --&gt; diff\u00e9rence = 10989\n10989 --&gt; 1899 , 99810 --&gt; diff\u00e9rence = 97911\n97911 --&gt; 11799 , 99711 --&gt; diff\u00e9rence = 87912\n87912 --&gt; 12789 , 98721 --&gt; diff\u00e9rence = 85932\n85932 --&gt; 23589 , 98532 --&gt; diff\u00e9rence = 74943 &lt;===\n74943 --&gt; 34479 , 97443 --&gt; diff\u00e9rence = 62964\n62964 --&gt; 24669 , 96642 --&gt; diff\u00e9rence = 71973\n71973 --&gt; 13779 , 97731 --&gt; diff\u00e9rence = 83952\n83952 --&gt; 23589 , 98532 --&gt; diff\u00e9rence = 74943 &lt;=== cycle<\/pre>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Modification_du_programme_Python\"><\/span>Modification du programme Python<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Plut\u00f4t que de d&rsquo;interrompre manuellement le programme, autant le modifier pour qu&rsquo;il s&rsquo;arr\u00eate de lui-m\u00eame quand une boucle est d\u00e9tect\u00e9e:<\/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=\"\">def d(n):\n    L = list( str(n) )\n    L.sort()\n    smallest = ''.join(L)\n    L.reverse()\n    biggest = ''.join( L )\n    return int(smallest) , int(biggest) , int(biggest)-int(smallest)\n\nn = int( input(\"Entrez un nombre entier : \") )\nant = 0\nN = [ n ]\nadd = False\n\nwhile not add:\n    print('{} --> {} , {} --> diff\u00e9rence = {}'.format(n , d(n)[0] , d(n)[1] , d(n)[2]))\n    n = d(n)[2]\n    if n not in N:\n        N += [ n ]\n    else:\n        add = True<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Ainsi, en entrant:$$n=78545$$on a:<\/p>\n\n\n\n<pre class=\"wp-block-preformatted\">78545 --&gt; 45578 , 87554 --&gt; diff\u00e9rence = 41976\n41976 --&gt; 14679 , 97641 --&gt; diff\u00e9rence = 82962\n82962 --&gt; 22689 , 98622 --&gt; diff\u00e9rence = 75933\n75933 --&gt; 33579 , 97533 --&gt; diff\u00e9rence = 63954\n63954 --&gt; 34569 , 96543 --&gt; diff\u00e9rence = 61974\n61974 --&gt; 14679 , 97641 --&gt; diff\u00e9rence = 82962<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Pour \\(n=55155\\), on a:<\/p>\n\n\n\n<pre class=\"wp-block-preformatted\">55155 --&gt; 15555 , 55551 --&gt; diff\u00e9rence = 39996\n39996 --&gt; 36999 , 99963 --&gt; diff\u00e9rence = 62964\n62964 --&gt; 24669 , 96642 --&gt; diff\u00e9rence = 71973\n71973 --&gt; 13779 , 97731 --&gt; diff\u00e9rence = 83952\n83952 --&gt; 23589 , 98532 --&gt; diff\u00e9rence = 74943\n74943 --&gt; 34479 , 97443 --&gt; diff\u00e9rence = 62964<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">(ah zut ! j&rsquo;avais promis d&rsquo;arr\u00eater&#8230;)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ce qu&rsquo;il y a de bien avec ce programme est que l&rsquo;on peut entrer un nombre entier quelconque, \u00e7a fonctionne!<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Pour en savoir un peu plus, vous pouvez regarder par exemple la page <a aria-label=\"undefined (s\u2019ouvre dans un nouvel onglet)\" href=\"https:\/\/www.dcode.fr\/algorithme-kaprekar\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/www.dcode.fr\/algorithme-kaprekar<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Nous allons parler dans cet article, inspir\u00e9 de la vid\u00e9o suivante, des constantes de Kaprekar et aller un peu plus loin.<\/p>\n","protected":false},"author":1,"featured_media":3727,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,5],"tags":[],"class_list":["post-2976","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","category-python"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Particularit\u00e9s num\u00e9riques (constantes de Kaprekar) - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Les constantes de Kaprekar sont obtenues par transformation d&#039;entiers selon une r\u00e8gle bien particuli\u00e8re. Regardons cela de plus pr\u00e8s.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Particularit\u00e9s num\u00e9riques (constantes de Kaprekar) - Mathweb.fr\" \/>\n<meta property=\"og:description\" content=\"Les constantes de Kaprekar sont obtenues par transformation d&#039;entiers selon une r\u00e8gle bien particuli\u00e8re. Regardons cela de plus pr\u00e8s.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathweb.fr\" \/>\n<meta property=\"article:published_time\" content=\"2020-07-23T14:34:20+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2020-10-03T13:20:54+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/07\/kaprekar-head.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"740\" \/>\n\t<meta property=\"og:image:height\" content=\"198\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"St\u00e9phane Pasquet\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u00c9crit par\" \/>\n\t<meta name=\"twitter:data1\" content=\"St\u00e9phane Pasquet\" \/>\n\t<meta name=\"twitter:label2\" content=\"Dur\u00e9e de lecture estim\u00e9e\" \/>\n\t<meta name=\"twitter:data2\" content=\"5 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2020\\\/07\\\/23\\\/particularites-numeriques-constantes-kaprekar\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2020\\\/07\\\/23\\\/particularites-numeriques-constantes-kaprekar\\\/\"},\"author\":{\"name\":\"St\u00e9phane Pasquet\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#\\\/schema\\\/person\\\/e4d3bb07968238378f0d5052a70dcd69\"},\"headline\":\"Particularit\u00e9s num\u00e9riques (constantes de Kaprekar)\",\"datePublished\":\"2020-07-23T14:34:20+00:00\",\"dateModified\":\"2020-10-03T13:20:54+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2020\\\/07\\\/23\\\/particularites-numeriques-constantes-kaprekar\\\/\"},\"wordCount\":594,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#\\\/schema\\\/person\\\/e4d3bb07968238378f0d5052a70dcd69\"},\"image\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2020\\\/07\\\/23\\\/particularites-numeriques-constantes-kaprekar\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2020\\\/07\\\/kaprekar-head.jpg\",\"articleSection\":[\"Math\u00e9matiques\",\"Python\"],\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2020\\\/07\\\/23\\\/particularites-numeriques-constantes-kaprekar\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2020\\\/07\\\/23\\\/particularites-numeriques-constantes-kaprekar\\\/\",\"url\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2020\\\/07\\\/23\\\/particularites-numeriques-constantes-kaprekar\\\/\",\"name\":\"Particularit\u00e9s num\u00e9riques (constantes de Kaprekar) - Mathweb.fr\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2020\\\/07\\\/23\\\/particularites-numeriques-constantes-kaprekar\\\/#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2020\\\/07\\\/23\\\/particularites-numeriques-constantes-kaprekar\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2020\\\/07\\\/kaprekar-head.jpg\",\"datePublished\":\"2020-07-23T14:34:20+00:00\",\"dateModified\":\"2020-10-03T13:20:54+00:00\",\"description\":\"Les constantes de Kaprekar sont obtenues par transformation d'entiers selon une r\u00e8gle bien particuli\u00e8re. Regardons cela de plus pr\u00e8s.\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2020\\\/07\\\/23\\\/particularites-numeriques-constantes-kaprekar\\\/#breadcrumb\"},\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2020\\\/07\\\/23\\\/particularites-numeriques-constantes-kaprekar\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"fr-FR\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2020\\\/07\\\/23\\\/particularites-numeriques-constantes-kaprekar\\\/#primaryimage\",\"url\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2020\\\/07\\\/kaprekar-head.jpg\",\"contentUrl\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2020\\\/07\\\/kaprekar-head.jpg\",\"width\":740,\"height\":198,\"caption\":\"constantes de kaprekar\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/2020\\\/07\\\/23\\\/particularites-numeriques-constantes-kaprekar\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Accueil\",\"item\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Particularit\u00e9s num\u00e9riques (constantes de Kaprekar)\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#website\",\"url\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/\",\"name\":\"Mathweb.fr\",\"description\":\"Math\u00e9matiques, LaTeX et Python\",\"publisher\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#\\\/schema\\\/person\\\/e4d3bb07968238378f0d5052a70dcd69\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"fr-FR\"},{\"@type\":[\"Person\",\"Organization\"],\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/#\\\/schema\\\/person\\\/e4d3bb07968238378f0d5052a70dcd69\",\"name\":\"St\u00e9phane Pasquet\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"fr-FR\",\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2025\\\/06\\\/cropped-logo-mathweb.webp\",\"url\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2025\\\/06\\\/cropped-logo-mathweb.webp\",\"contentUrl\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2025\\\/06\\\/cropped-logo-mathweb.webp\",\"width\":74,\"height\":77,\"caption\":\"St\u00e9phane Pasquet\"},\"logo\":{\"@id\":\"https:\\\/\\\/www.mathweb.fr\\\/euclide\\\/wp-content\\\/uploads\\\/2025\\\/06\\\/cropped-logo-mathweb.webp\"}}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Particularit\u00e9s num\u00e9riques (constantes de Kaprekar) - Mathweb.fr","description":"Les constantes de Kaprekar sont obtenues par transformation d'entiers selon une r\u00e8gle bien particuli\u00e8re. Regardons cela de plus pr\u00e8s.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/","og_locale":"fr_FR","og_type":"article","og_title":"Particularit\u00e9s num\u00e9riques (constantes de Kaprekar) - Mathweb.fr","og_description":"Les constantes de Kaprekar sont obtenues par transformation d'entiers selon une r\u00e8gle bien particuli\u00e8re. Regardons cela de plus pr\u00e8s.","og_url":"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/","og_site_name":"Mathweb.fr","article_published_time":"2020-07-23T14:34:20+00:00","article_modified_time":"2020-10-03T13:20:54+00:00","og_image":[{"width":740,"height":198,"url":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/07\/kaprekar-head.jpg","type":"image\/jpeg"}],"author":"St\u00e9phane Pasquet","twitter_card":"summary_large_image","twitter_misc":{"\u00c9crit par":"St\u00e9phane Pasquet","Dur\u00e9e de lecture estim\u00e9e":"5 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/#article","isPartOf":{"@id":"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/"},"author":{"name":"St\u00e9phane Pasquet","@id":"https:\/\/www.mathweb.fr\/euclide\/#\/schema\/person\/e4d3bb07968238378f0d5052a70dcd69"},"headline":"Particularit\u00e9s num\u00e9riques (constantes de Kaprekar)","datePublished":"2020-07-23T14:34:20+00:00","dateModified":"2020-10-03T13:20:54+00:00","mainEntityOfPage":{"@id":"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/"},"wordCount":594,"commentCount":0,"publisher":{"@id":"https:\/\/www.mathweb.fr\/euclide\/#\/schema\/person\/e4d3bb07968238378f0d5052a70dcd69"},"image":{"@id":"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/#primaryimage"},"thumbnailUrl":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/07\/kaprekar-head.jpg","articleSection":["Math\u00e9matiques","Python"],"inLanguage":"fr-FR","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/","url":"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/","name":"Particularit\u00e9s num\u00e9riques (constantes de Kaprekar) - Mathweb.fr","isPartOf":{"@id":"https:\/\/www.mathweb.fr\/euclide\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/#primaryimage"},"image":{"@id":"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/#primaryimage"},"thumbnailUrl":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/07\/kaprekar-head.jpg","datePublished":"2020-07-23T14:34:20+00:00","dateModified":"2020-10-03T13:20:54+00:00","description":"Les constantes de Kaprekar sont obtenues par transformation d'entiers selon une r\u00e8gle bien particuli\u00e8re. Regardons cela de plus pr\u00e8s.","breadcrumb":{"@id":"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/#breadcrumb"},"inLanguage":"fr-FR","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/"]}]},{"@type":"ImageObject","inLanguage":"fr-FR","@id":"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/#primaryimage","url":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/07\/kaprekar-head.jpg","contentUrl":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/07\/kaprekar-head.jpg","width":740,"height":198,"caption":"constantes de kaprekar"},{"@type":"BreadcrumbList","@id":"https:\/\/www.mathweb.fr\/euclide\/2020\/07\/23\/particularites-numeriques-constantes-kaprekar\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Accueil","item":"https:\/\/www.mathweb.fr\/euclide\/"},{"@type":"ListItem","position":2,"name":"Particularit\u00e9s num\u00e9riques (constantes de Kaprekar)"}]},{"@type":"WebSite","@id":"https:\/\/www.mathweb.fr\/euclide\/#website","url":"https:\/\/www.mathweb.fr\/euclide\/","name":"Mathweb.fr","description":"Math\u00e9matiques, LaTeX et Python","publisher":{"@id":"https:\/\/www.mathweb.fr\/euclide\/#\/schema\/person\/e4d3bb07968238378f0d5052a70dcd69"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/www.mathweb.fr\/euclide\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"fr-FR"},{"@type":["Person","Organization"],"@id":"https:\/\/www.mathweb.fr\/euclide\/#\/schema\/person\/e4d3bb07968238378f0d5052a70dcd69","name":"St\u00e9phane Pasquet","image":{"@type":"ImageObject","inLanguage":"fr-FR","@id":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/06\/cropped-logo-mathweb.webp","url":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/06\/cropped-logo-mathweb.webp","contentUrl":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/06\/cropped-logo-mathweb.webp","width":74,"height":77,"caption":"St\u00e9phane Pasquet"},"logo":{"@id":"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/06\/cropped-logo-mathweb.webp"}}]}},"_links":{"self":[{"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/posts\/2976","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/comments?post=2976"}],"version-history":[{"count":0,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/posts\/2976\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/media\/3727"}],"wp:attachment":[{"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/media?parent=2976"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/categories?post=2976"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.mathweb.fr\/euclide\/wp-json\/wp\/v2\/tags?post=2976"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}