10 different digits, 9 products

> On Tue, Jan 3, 2012 at 11:08 PM, Eric Angelini wrote:

>

> Hello SeqFans,

> I’m looking for all D numbers with 10 digits (digits must be

> different one from another) having this property:

> When you multiply two touching digits of D, the result is

> visible in D (as a character substring).

>

> Example:

> 9071532486 --> the product 9*0 is in D ("0"); the product 0*7

>is in D too ("0"); 7*1 ("7"); 1*5 ("5"); 5*3 ("15"); 3*2 ("6");

> 2*4 ("8"); 4*8 ("32") and 8*6 ("48").

> I have also 4297631805 (4*2="8"; 2*9="18"; 9*7="63"; 7*6="42";

> 6*3="18"; etc.)

> The same for 5420976318, 7963205418, 5630187924, 5678142309...

> This has a flavor of Zak’s http://oeis.org/A134962

> Best,

> É.

Jason Kimberley was quick to answer:

Here are my 58 solutions (found by brute force over 10!):

3207154869, 3205486917, 4063297185, 4063792185, 4230567819, 4230915678, 4297630518, 4297631805, 5042976318, 5063297184, 5079246318, 5093271486, 5094236718, 5148609327, 5180429763, 5180792463, 5180942367, 5184063297, 5420796318, 5420976318, 5486913207, 5630187924, 5630241879, 5630418792, 5630421879, 5630429187, 5630429718, 5630792418, 5630924187, 5678142309, 6320184597, 6320459718, 6320718459, 6320791845, 6320971845, 6324079185, 6324097185, 6329705184, 6329718405, 7091532486, 7132054869, 7153248609, 7183092465, 7924063185, 7924630518, 7924631805, 7963205418, 9071532486, 9142305678, 9153248607, 9246518307, 9305614278, 9308142765, 9327051486, 9327148605, 9423670518, 9423671805, 9872305614.

I have started creating this sequence as https://oeis.org/draft/A198298

Many thanks, Jason!

The generalization by Maximilian Hasler came quickly too:

Hello Eric,

I have created the sequence http://oeis.org/A203569:

Numbers whose digits are a permutation of [0,...,n] and which contain the product of any two adjacent digits as a substring.

This generalizes the problem to a smaller number of digits.

I found it interesting that considering permutations of [1,...,n] instead yields only three nontrivial solutions, 3412, 4312 and 71532486 (plus the trivial solutions 1, 12, 21, 213, 312).

Another interesting type of generalization, also computed by Maximilian: http://oeis.org/A203566.

Jean-Marc Falcoz has computed all 10-digit such D numbers where we want respectively 3 and 4 “touching digits” to be multiplied one by the others and have the result visible in D:

Voici les solutions avec 3 multiplications :

5631890724, 6581324079, 6581324097, 7249056318.

Et avec 4 :

4327059168, 4613590728, 4613590872, 7860241359, 7860291354, 8490536127, 8760241359, 8760291354.

(Example for 4327059168: 4*3*2*7= “168”; 5*9*1*6=“270”; 9*1*6*8=“432”)

Then came another idea by Jean-Paul Davalan:

Autre proposition de suite : on assortit chiffres et bases.

Le nombre N de n chiffres sera écrit "abc...d" dans la base n, ses n chiffres a, b, c, ..., d seront différents et tous les produits ab, bc, ... de deux chiffres consécutifs, écrits aussi en base n, figureront dans l’écriture "abc...d" de N.

Exemple : 5321604 a 7 chiffres et est écrit dans la base 7, ses deux premiers chiffres sont 5 et 3 leur produit est 21 en base 7. Ce 21 se retrouve dans le nombre.

Cette fois la suite a de bonnes chances d’être infinie.

(Mais comment écrire tous ces nombres dans l’OEIS, en particulier lorsque la base est plus grande que 10 ?)

(base 2) 10

(base 3) 102 120 201 210

(base 4) 1203 1230 1302 2013 2031 2103 2130 3012 3021 3102 3120

(base 5) 13024 13042 20314 20413 21304 21403 24013 24130 30214 30412 31204 31402 40213 40312 41203 41302 42013 42130

(base 6) 203415 204315 205134 205143 235104 250314 251403 302514 305124 305214 312405 314025 314052 321045 321054 341205 410235 431205 451032 503124 512034 512043 512403 520143 520314 521403 541032

(base 7) 1504326 1506234 1540326 1543026 1543260 2153406 2340615 2341506 2601543 2603154 2603415 2604315 2615034 2615043 2615403 2615430 3026154 3154026 3260154 3260415 3261504 3261540 3402615 3406215 3415026 3415062 4032615 4053216 4061325 4062153 4062315 4132506 4150326 4150623 4302615 4306215 4315026 4315062 4320615 4321506 4326015 4326150 5321406 5321604 6021534 6023415 6041325 6043215 6053214 6132504 6150234 6150432 6203415 6204315 6215034 6215043 6215304 6215340 6230415 6231504 6234015 6234150

(base 8) 14326057 14326075 26057143 26075143 26143057 26143075 50714326 51432607 51462307 57014326 57026143 57061432 57062143 57143026 57143062 57143206 57143260 60571432 60751432 61432057 61432075 62057143 62075143 62143057 62143075 70514326 71432605 71462305 72305164 73046125 73064125 75014326 75026143 75061432 75062143 75143026 75143062 75143206 75143260

(base 9) 247053168 264057138 264075138 264138057 264138075 426057138 426075138 426138057 426138075 531680247 531680742 532407168 532416807 570264138 570426138 571380264 571380426 705324168 708423516 716084235 716805324 742053168 750264138 750426138 751380264 751380426 842071635 842350716 842351607

(base 10 – cf. Jason) 3205486917 3207154869 4063297185 4063792185 4230567819 4230915678 4297630518 4297631805 5042976318 5063297184 5079246318 5093271486 5094236718 5148609327 5180429763 5180792463 5180942367 5184063297 5420796318 5420976318 5486913207 5630187924 5630241879 5630418792 5630421879 5630429187 5630429718 5630792418 5630924187 5678142309 6320184597 6320459718 6320718459 6320791845 6320971845 6324079185 6324097185 6329705184 6329718405 7091532486 7132054869 7153248609 7183092465 7924063185 7924630518 7924631805 7963205418 9071532486 9142305678 9153248607 9246518307 9305614278 9308142765 9327051486 9327148605 9423670518 9423671805 9872305614

...

Many thanks to all,

É.

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French version:

Je cherche tous les nombres D de 10 chiffres, (chiffres devant être tous différents), qui ont la propriété suivante :

quand on multiplie deux chiffres qui se touchent dans D, le résultat est visible dans D (comme sous-chaîne de caractères).

Exemple, 9071532486.

Le produit 9*0 est dans D ("0"); le produit 0*7 aussi ("0"); 7*1 ("7"); 1*5 ("5"); 5*3 ("15"); 3*2 ("6"); 2*4 ("8"); 4*8 ("32") et 8*6 ("48").

J’ai aussi 4297631805 (4*2="8"; 2*9="18"; 9*7="63"; 7*6="42"; 6*3="18"; etc.

Pareil pour 5420976318, 7963205418, 5630187924, 5678142309...

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More “Sweet” sequences here.