Two factors
(deux facteurs)
Nombres N décomposables en deux chaînes de
caractères pouvant être vues comme des facteurs de ce même nombre N. Ainsi 315 sera-t-il
lu comme égal à 3.15.k (avec k valant 7 ici --> 315 = 3.15.7)
Le nombre 42336 fait aussi partie de la
suite S ci-dessous car il peut être décomposé en 42.336.k (avec k = 3).
Merci à Alois
Heinz, Hans Havermann et Jean-Marc Falcoz qui ont tous trois
calculé indépendamment la suite S ci-dessous.
[Le problème avait été publié sur la liste SeqFan le 16 mars 2012]
> Hello SeqFan,
> looking for integers a, b and c such that a.b.c = [ab]
> [ab] being the concatenation of a and b.
>
> An S sequence of such [ab]s would include those:
>
> S = 11, 12, 15, 24, 36, 110, 120, 150, 240, 360, ...
> a = 1 1 1 2 3 1 1 1 2 3 ...
> b = 1 2 5 4 6 10 20 50 40 60 ...
> c = 11 6 3 3 2 11 6 3 3 2 ...
>
> Are there [ab] integers escaping the above pattern?
>
> In graphic terms, we are looking for integers
> like ABCDEFGHIJKLM divible by the product, say,
> ABC.DEFGHIJKLM or ABCDEF.GHIJKLM or ABCDEFGHIJKL.M
> or... etc.
> (With ABC = 360 we have ABC divisible by A.BC)
>
> Best,
> É.
S = 11, 12, 15, 24, 36, 110, 120, 125, 150, 240, 315, 360, 735, 1100,
1125, 1200, 1250, 1352, 1500, 1734, 2400, 3150, 3375, 3600, 7350, 11000, 11250,
12000, 12500, 13520, 14112, 15000, 17340, 18144, 21168, 24000, 31500, 33750, 36000,
42336, 63504, 67335, 73500, 91125, 110000, 112500, 120000, 125000, 135200, 141120,
143143, 150000, 167334, 173400, 181440, 211680, 240000, 273375, 315000, 337500,
360000, 423360, 635040, 673350, 735000, 911250, 1100000, 1125000, 1134375, 1200000,
1250000, 1352000, 1391112, 1411200, 1431430, 1500000, 1673340, 1734000, 1814400,
2116800, 2314375, 2400000, 2733750, 3135008, 3150000, 3375000, 3600000, 4173336,
4233600, 6350400, 6673335, 6733500, 7350000, 8911125, 9112500, 11000000, 11250000,
11343750, 12000000, 12500000, 13203125, 13520000, 13911120, 14112000, 14314300,
15000000, 16673334, 16733400, 17340000, 18144000, 21168000, 23143750, 24000000,
26733375, 27337500, 31350080, 31500000, 33750000, 36000000, 41733360, 42336000,
52116672, 63504000, 66733350, 67335000, 73500000, 89111250, 89314288, 91125000,
...
Voici la même suite avec un point séparant
les deux facteurs :
S = 1.1, 1.2, 1.5, 2.4, 3.6, 1.10, 1.20, 1.25, 1.50, 2.40, 3.15, 3.60,
7.35, 1.100, 1.125, 1.200, 1.250, 13.52, 1.500, 17.34, 2.400, 3.150, 3.375, 3.600,
7.350, 1.1000, 1.1250, 1.2000, 1.2500, 13.520, 14.112, 1.5000, 17.340, 18.144, 21.168,
2.4000, 3.1500, 3.3750, 3.6000, 42.336, 63.504, 67.335, 7.3500, 9.1125, 1.10000,
1.12500, 1.20000, 1.25000, 13.5200, 14.1120, 143.143, 1.50000, 167.334, 17.3400,
18.1440, 21.1680, 2.40000, 27.3375, 3.15000, 3.37500, 3.60000, 42.3360, 63.5040,
67.3350, 7.35000, 9.11250, 1.100000, 1.125000, 11.34375, 1.200000, 1.250000, 13.52000,
139.1112, 14.11200, 143.1430, 1.500000, 167.3340, 17.34000, 18.14400, 21.16800,
23.14375, 2.400000, 27.33750, 313.5008, 3.150000, 3.375000, 3.600000, 417.3336,
42.33600, 63.50400, 667.3335, 67.33500, 7.350000, 89.11125, 9.112500, 1.1000000,
1.1250000, 11.343750, 1.2000000, 1.2500000, 13.203125, 13.520000, 139.11120, 14.112000,
143.14300, 1.5000000, 1667.3334, 167.33400, 17.340000, 18.144000, 21.168000, 23.143750,
2.4000000, 267.33375, 27.337500, 313.50080, 3.1500000, 3.3750000, 3.6000000, 417.33360,
42.336000, 521.16672, 63.504000, 667.33350, 67.335000, 7.3500000, 89.111250, 893.14288,
9.1125000, ...
Merci à tous,
Best,
É.