J’ai posé fin mai 2007 à quelques amis et aux listes Seq-Fans, Math-Fun et fr.sci.maths la question suivante :

> Pourriez-vous m'aider et me dire si la suite T ci-dessous est finie ?

> T = 1,2,3,4,5,6,7,8,9,11,13,14,15,17,18,19,21,22,23,25, ...

> Elle est strictement croissante, commence par 1, cherche à être prolongée par le plus petit terme suivant possible et répond à la loi :

--> la somme S de deux termes consécutifs de la suite ne partage aucun chiffre avec les termes d'origine.

> Ainsi:

1+2=3 (et "3" ne partage aucun chiffre ni avec "1" ni avec "2")

2+3=5 (et "5" ne partage aucun chiffre ni avec "2" ni avec "3")

3+4=7 (et "7" ne partage aucun chiffre ni avec "3" ni avec "4")

...

9+11=20 (et "20" ne partage aucun chiffre ni avec "9" ni avec "11")

--> on voit que "10" n'a pas sa place dans la suite, après "9", car :

9+10=19 et "19" partage un chiffre au moins avec l'un des termes

(ici "19" partage à la fois "1" avec "10" et "9" avec "9")

--> de même "16" est-il interdit après "15" car :

15+16=31 et 31 partage "1" avec 15 (et avec 16).

> etc.

T (calculée par « Lapinot » sur fr.sci.maths) commence comme ceci :

1,2,3,4,5,6,7,8,9,11,13,14,15,17,18,19,21,22,23,25,35,36,38,39,41,42,43,44,45,46,47,48,51,149,151,152,154,155,165,167,168,169,171,172,176,177,178,181,182,184,185,188,189,211,219,221,222,223,225,235,236,238,239,261,269,271,272,273,276,277,278,281,282,283,284,286,288,289,311,313,314,315,325,335,339,341,344,345,347,348,351,353,354,355,356,358,359,361,363,364,365,382,383,384,385,386,388,389,622,678,722,723,725,735,753,847,848,852,854,855,856,858,859,863,864,865,866,868,869,882,883,884,885,886,888,889,1111,1113,1114,1115,1117,1118,1119,1131,1133,1134,1135,1139,1141,1144,1145,1147,1148,1151,1155,1165,1167,1168,1169,1171,1175,1185,1188,1189,1311,1313,1314,1315,1331,1333,1334,1335,1339,1341,1344,1345,1347,1348,1351,1353,1354,1355,1365,1413,1414,1415,1417,1419,1431,1434,1435,1441,1444,1445,1447,1451,1455,1465,1467,1468,1471,1481,1484,1485,1488,1504,1718,1721,1722,1726,1727,1728,1762,1768,1771,1772,1776,1777,1778,1781,1782,1784,1808,1814,1815,1817,1818,1819,1821,1822,1825,1841,1851,1852,1854,1855,1865,1868,1869,1881,1882,1884,1885,1888,1889,2111,2189,2211,2219,2221,2222,2223,2225,2235,2236,2238,2239,2261,2269,2271,2272,2273,2276,2277,2278,2281,2282,2283,2288,2289,2311,2329,2331,2333,2335,2339,2361,2383,2388,2389,2611,2689,2711,2719,2721,2722,2723,2726,2727,2728,2731,2733,2761,2769,2771,2772,2773,2776,2777,2778,2781,2782,2783,2808,2809,2822,2823,2827,2828,2829,2831,2833,2834,2841,2862,2868,2869,2881,2882,2883,2884,2886,2888,2889,3111,3113,3114,3115,3131,3133,3134,3135,3139,3141,3144,3145,3147,3148,3151,3253,3347,3348,3351,3353,3354,3355,3359,3381,3383,3384,3385,3389,3611,3613,3614,3615,3633,3638,3639,3641,3644,3645,3646,3651,3819,3821,3823,3828,3829,3831,3833,3834,3835,3839,3861,3863,3864,3865,3882,3883,3884,3885,3886,3888,3889,6222,6778,7222,7223,7225,7235,7253,7353,7509,8703,13519,13541,13545,13551,13649,13903,13919,13931,13933,13934,13936,13939,13941,13944,14111,14114,14115,14117,14119,14131,14134,14135,14141,14144,14145,14147,14151,14155,14165,14167,14171,14179,14341,14344,14345,14347,14351,14355,14365,14413,14414,14415,14417,14419,14431,14434,14435,14441,14444,14445,14447,14451,14455,14465,14467,14471,14511,14515,14517,14543,14544,14545,14547,14551,14651,14656,14664,14665,14667,14668,14671,14681,14711,14811,14814,14815,14817,14818,14834,14835,14841,14851,14855,14865,14868,14884,14885,14888,15034,47188,47812,47818,47821,47834,48077,48078,48083,48084,48085,48088,48134,48135,48141,48151,48152,48154,48155,48171,48181,48182,48184,48185,48188,48205,48408,48503,48608,48614,48615,48618,48634,48636,48641,48651,48652,48655,48656,48664,48665,48666,48684,48686,48814,48815,48818,48821,48834,48835,48841,48851,48852,48854,48855,48856,48863,48864,48865,48866,48884,48885,48886,50225,60886,60888,60889,63333,63668,63803,63908,77203,77208,77226,77227,77228,77232,77233,77262,77268,77272,77273,77276,77277,77278,77282,77283,77308,77333,77608,77633,77647,77648,77662,77668,77672,77676,77677,77678,77722,77723,77726,77727,77728,77732,77733,77762,77768,77772,77773,77776,77777,77778,77782,77783,77808,77827,77828,77832,77833,78078,78083,78328,78332,78333,80778,80783,80808,80809,80822,80823,80828,80829,80832,80833,80834,80835,80839,80883,80884,80885,80888,80889,82222,82223,82227,82228,82229,82232,82233,82238,82239,82272,82273,82277,82278,82282,82283,82288,82289,82322,82323,82327,82328,82329,82332,82333,82338,82339,82722,82723,82727,82728,82732,82733,82772,82773,82777,82778,82782,82783,82808,82809,82822,82823,82827,82828,82829,82832,82833,82834,82842,82848,82882,82883,82884,82889,83222,83223,83227,83228,83229,83232,83233,83238,83239,83272,83273,83277,83278,83282,83283,83284,83292,83322,83323,83327,83328,83329,83332,83333,83334,83335,83339,83383,83384,83385,83389,83833,83834,83835,83839,83883,83884,83885,83889,86222,86822,86882,86888,86889,88222,88223,88228,88229,88232,88233,88238,88239,88302,88309,88332,88333,88334,88335,88339,88383,88384,88385,88389,88622,88682,88688,88689,88822,88823,88828,88829,88832,88833,88834,88835,88839,88862,88868,88869,88882,88883,88884,88885,88886,88888,88889,111111,111113,111114,111115,111117,111118,111119,111131,111133,111134,111135,111139,111141,111144,111145,111147,111148,111151,111155,111165,111167,111168,111169,111171,111175,111185,111188,111189,111311,111313,111314,111315,111331,111333,111334,111335,111339,111341,111344,111345,111347,111348,111351,111353,111354,111355,111365,111413,111414,111415,111417,111419,111431,111434,111435,111441,111444,111445,111447,111451,111455,111465,111467,111468,111471,111481,111484,111485,111488,111504,111718,111751,111849,111851,111855,111865,111868,111869,111881,111884,111885,111888,111889,113111,113113,113114,113115,113131,113133,113134,113135,113139,113141,113144,113145,113147,113148,113151,113311,113313,113314,113315,113331,113333,113334,113335,113339,113341,113344,113345,113347,113348,113351,113353,113354,113355,113385,113389,113611,113613,113614,113615,113633,113639,113641,113644,113645,113653,113819,113831,113833,113834,113835,113839,113861,113863,113864,113865,113909,1139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Nicolas Graner m’a fait quelques remarques :

> Il y a évidemment des nombres qui n'ont pas de successeur, par exemple 123456789, ou tous ceux qui se terminent par 0, ou 98. Si une suite

tombe un jour sur un de ces nombres, elle est finie. Mais le raisonnement qui montre que ces nombres n'ont pas de successeur marche aussi à l'envers, et montre aussi qu'ils n'ont pas de prédécesseur. Ainsi, la seule suite qui contient 98 est celle qui commence par 98, et elle n'a qu'un terme. Pas très intéressant.

Plus intéressant, on peut trouver des suites de deux termes n'ayant ni prédécesseur ni successeur. Exemple : en partant de 203040506070809, le terme suivant est 908070605040302, et ça s'arrête là.

(...)

"Bermisch" a écrit dans le même sens sur fr.sci.maths le 30 mai :

> Hé, un M de la suite ne peut pas contenir les 10 chiffres par construction (pas de prédécesseur), mais également : si un M de la suite contient tous les chiffres de 1 à 9, et pas 0, pour tout N>M, S=M+N contient d'autres chiffres que 0 qui est le seul permis, la suite s'arrête là. Heureusement, un N de ce type n'a pas non plus de M prédécesseur :-). On voit aussi que la suite ne contient pas de N multiple de 10 sinon S=M+N aurait même chiffre des unités que M. Ceci ferait penser à traiter le pb dans les anneaux d'entiers modulo 10^k où une telle suite a un comportement plus facile à maitriser, avant de passer ad infinitum. On peut aussi essayer avec une autre base que 10, en commençant par 2 (facile, non, la suite s'arrête à "1" et c'est tout :-) avant d'essayer de généraliser.

-----

Notons que si l’on ne s’oblige pas à prolonger la suite T par « le plus petit entier disponible », la réponse est évidente : oui, une telle suite peut être infinie, pour peu qu’on choisisse bien les termes à additionner. Ainsi Nicolas Graner propose-t-il encore :

T = 6, 66, 666, 6666, 66666, 66666, 6666666, 666...666, où les sommes successives valent 72, 732, 7332, 73332, 733332, 7333332, 7333...3332, etc. Elles ne comporteront jamais de « 6 ».

Une suite de ce genre « plus lente » pourrait être celle-ci :

T = 1, 2, 28, 272, 2728, 27272, 272728, 2727272, 27272728,... où la somme de deux termes consécutifs quelconques ne partage jamais de chiffre avec lesdits termes, comme on peut le voir ici :

1+2=3

2+28=30

28+272=300

272+2728=3000

2728+27272=30000

27272+272728=300000

...

On comprend la méthode de construction (fabriquer le complémentaire à 300...000 chaque fois à partir du dernier terme de T)

Voici un autre exemple :

T = 1, 5, 55, 545, 5455, 54545, 545455, 5454545, ... avec les sommes :

1+5=6

5+55=60

55+545=600

545+5455=6000

5455+54545=60000

54545+545455=600000

etc.

Olivier Miakinen a proposé ceci (toujours sur fr.sci.maths) :

> Si on retire le critère selon lequel le terme suivant doit être le plus

petit possible, alors on peut à la fois construire des suites infinies

et d'autres qui ont un dernier terme à partir duquel on ne peut pas en

rajouter.

Voici d'abord un exemple de suite infinie :

1

        2

        9

       11

       22

       93

      111

      222

      933

     1111

     2222

     9333

    11111

    22222

    93333

      ...

> Pour une suite finie, je pars du 111111111 dont je sais d'après ma suite

infinie que je peux l'obtenir, et je fais :

111111111

331393339

339112349

368595421

409182356

À partir de 409182356 il n'y a plus de continuation possible, puisque la

seule somme possible ne contenant que des 7 est le nombre précédent, qui

est plus petit.

Bien vu, merci Olivier !

Il a été prouvé ici que la suite commençant par 0, 1, 2, dont la somme de *trois* termes consécutifs ne peut jamais comporter de chiffre figurant dans l’un des trois termes à additionner, est finie.

À suivre...

__________

Retour à la page d’accueil ? Additionner ceci et ça, là.