Weird Addition

Hello SeqFans,

Let’s write down an AB integer (2 digits, A and B) using a square per digit and decide to write the result of the addition of A+B on the first ‘x’ mark, or on the two ‘x’ marks if needed, like this (again, one digit per square):

A B

. x

So 10 produces 10

And 19 produces 19

Now, sometimes we are forced to repeat the addition, as a new integer appears, having more than one digit:

123 produces 35 and 35 produces 8: 123

.35

..8

193 produces this ‘array’: 193

.11

.02

..2

But 192 and 194, for instance, fail: 192 194

.11 .11

.01 .01

... as 11 shoud produce 11

.2 <--- and not .1 (like above)

The sequence of "failing integers" starts with 128, I guess:

128

.31

..0 <-- the zero, here, should be a 4, resulting from 3+1).

Could someone compute a few hundred terms of this "failing integers" sequence (if of some interest)?

Puzzle:

What is the biggest integer _having no zero_ that is NOT failing?

I have 5121212:

5121212

.633333

..96666

...1111

...5222

....744

.....18

.....19

......1

......0

Best,

É.

Jean-Marc Falcoz was quick to answer – and submitted me the full finite sequence of “NOT failing” numbers greater than 100.

This sequence ends with 5121212 as conjectured above:

> J’ai programmé ces weird additions (enfin, je crois, je te laisse vérifier, mais ça colle avec les exemples et contre-exemples que tu donnes). Je suis déçu, car je n’ai pas trouvé plus que 5121212 (je suis allé jusqu’à plus de 385 000 000). Le plus grand qui n’a pas de répétitions (dans le genre .121212) est 814325. Pour que tu puisses vérifier, je te copie la liste des entiers >100, et sans zéros, qui marchent :

{111, 112, 113, 114, 115, 116, 117, 118, 121, 122, 123, 124, 125, 126, 127, 131, 132, 133, 134, 135, 136, 141, 142, 143, 144, 145, 151, 152, 153, 154, 161, 162, 163, 171, 172, 181, 183, 193, 211, 212, 213, 214, 215, 216, 217, 218, 221, 222, 223, 224, 225, 226, 227, 231, 232, 233, 234, 235, 236, 241, 242, 243, 244, 245, 251, 252, 253, 254, 261, 262, 263, 271, 272, 274, 284, 293, 311, 312, 313, 314, 315, 316, 317, 318, 321, 322, 323, 324, 325, 326, 327, 331, 332, 333, 334, 335, 336, 341, 342, 343, 344, 345, 351, 352, 353, 354, 361, 362, 363, 365, 371, 375, 384, 393, 411, 412, 413, 414, 415, 416, 417, 418, 421, 422, 423, 424, 425, 426, 427, 431, 432, 433, 434, 435, 436, 441, 442, 443, 444, 445, 451, 452, 453, 454, 456, 461, 462, 466, 471, 475, 484, 493, 511, 512, 513, 514, 515, 516, 517, 518, 521, 522, 523, 524, 525, 526, 527, 531, 532, 533, 534, 535, 536, 541, 542, 543, 544, 545, 547, 551, 552, 553, 557, 561, 562, 566, 571, 575, 584, 593, 611, 612, 613, 614, 615, 616, 617, 618, 621, 622, 623, 624, 625, 626, 627, 631, 632, 633, 634, 635, 636, 638, 641, 642, 643, 644, 648, 651, 652, 653, 657, 661, 662, 666, 671, 675, 684, 693, 711, 712, 713, 714, 715, 716, 717, 718, 721, 722, 723, 724, 725, 726, 727, 729, 731, 732, 733, 734, 735, 739, 741, 742, 743, 744, 748, 751, 752, 753, 757, 761, 762, 766, 771, 775, 784, 793, 811, 812, 813, 814, 815, 816, 817, 818, 821, 822, 823, 824, 825, 826, 831, 832, 833, 834, 835, 839, 841, 842, 843, 844, 848, 851, 852, 853, 857, 861, 862, 866, 871, 875, 884, 893, 911, 912, 913, 914, 915, 916, 917, 921, 922, 923, 924, 925, 926, 931, 932, 933, 934, 935, 939, 941, 942, 943, 944, 948, 951, 952, 953, 957, 961, 962, 966, 971, 975, 984, 993, 1111, 1112, 1113, 1114, 1115, 1116, 1121, 1122, 1123, 1124, 1131, 1132, 1211, 1212, 1213, 1214, 1215, 1221, 1222, 1223, 1231, 1241, 1311, 1312, 1313, 1314, 1321, 1322, 1324, 1333, 1341, 1411, 1412, 1416, 1424, 1432, 1511, 1515, 1523, 1531, 1614, 1622, 1713, 1721, 1812, 1831, 1832, 1833, 1834, 1839, 1931, 1932, 1933, 1939, 2111, 2112, 2113, 2114, 2115, 2116, 2121, 2122, 2123, 2124, 2131, 2132, 2211, 2212, 2213, 2214, 2215, 2221, 2222, 2223, 2231, 2233, 2242, 2311, 2312, 2313, 2314, 2316, 2321, 2325, 2333, 2341, 2411, 2412, 2416, 2424, 2432, 2511, 2515, 2523, 2531, 2614, 2622, 2713, 2721, 2741, 2742, 2743, 2748, 2841, 2842, 2848, 2931, 2932, 2933, 2939, 3111, 3112, 3113, 3114, 3115, 3116, 3121, 3122, 3123, 3124, 3131, 3132, 3142, 3151, 3211, 3212, 3213, 3214, 3215, 3221, 3222, 3223, 3225, 3234, 3242, 3311, 3312, 3313, 3317, 3321, 3325, 3333, 3341, 3411, 3412, 3416, 3424, 3432, 3511, 3515, 3523, 3531, 3614, 3622, 3651, 3652, 3657, 3751, 3757, 3841, 3842, 3848, 3931, 3932, 3933, 3939, 4111, 4112, 4113, 4114, 4115, 4116, 4121, 4122, 4123, 4124, 4131, 4132, 4134, 4143, 4151, 4211, 4212, 4213, 4214, 4215, 4217, 4221, 4222, 4226, 4234, 4242, 4311, 4312, 4313, 4317, 4321, 4325, 4333, 4341, 4411, 4412, 4416, 4424, 4432, 4511, 4515, 4523, 4531, 4561, 4566, 4666, 4751, 4757, 4841, 4842, 4848, 4931, 4932, 4933, 4939, 5111, 5112, 5113, 5114, 5115, 5116, 5121, 5122, 5123, 5124, 5126, 5131, 5135, 5143, 5151, 5211, 5212, 5213, 5214, 5218, 5221, 5222, 5226, 5234, 5242, 5311, 5312, 5313, 5317, 5321, 5325, 5333, 5341, 5411, 5412, 5416, 5424, 5432, 5475, 5575, 5666, 5751, 5757, 5841, 5842, 5848, 5931, 5932, 5933, 5939, 6111, 6112, 6113, 6114, 6115, 6116, 6118, 6121, 6122, 6123, 6127, 6131, 6135, 6143, 6151, 6211, 6212, 6213, 6214, 6218, 6221, 6222, 6226, 6234, 6242, 6311, 6312, 6313, 6317, 6321, 6325, 6333, 6341, 6384, 6484, 6575, 6666, 6751, 6757, 6841, 6842, 6848, 6931, 6932, 6933, 6939, 7111, 7112, 7113, 7114, 7115, 7121, 7122, 7123, 7127, 7131, 7135, 7143, 7151, 7211, 7212, 7213, 7214, 7218, 7221, 7222, 7226, 7234, 7242, 7293, 7393, 7484, 7575, 7666, 7751, 7757, 7841, 7842, 7848, 7931, 7932, 7933, 7939, 8111, 8112, 8113, 8114, 8115, 8121, 8122, 8123, 8127, 8131, 8135, 8143, 8151, 8211, 8393, 8484, 8575, 8666, 8751, 8757, 8841, 8842, 8848, 8931, 8932, 8933, 8935, 8939, 9111, 9112, 9113, 9211, 9393, 9484, 9575, 9666, 9751, 9757, 9841, 9842, 9844, 9848, 9931, 9932, 9939, 11111, 11112, 11121, 11212, 12114, 12121, 12211, 12411, 12416, 13113, 13242, 13333, 13416, 14242, 14325, 15151, 15234, 15311, 15317, 16143, 16226, 17135, 17211, 17212, 17214, 17218, 18127, 18393, 19393, 21111, 21112, 21114, 21122, 21212, 22114, 22121, 22211, 22333, 22424, 23113, 23333, 23416, 24242, 24325, 25151, 25234, 25311, 25317, 26143, 26226, 27131, 27135, 27211, 27218, 27484, 28484, 29393, 31111, 31115, 31122, 31212, 31424, 31515, 32114, 32121, 32211, 32341, 32424, 33333, 33416, 34242, 34325, 35151, 35234, 35311, 35317, 36143, 36222, 36226, 36575, 37575, 38484, 39393, 41111, 41115, 41122, 41212, 41341, 41432, 41515, 42114, 42121, 42341, 42424, 43333, 43416, 44242, 44325, 45151, 45234, 45311, 45313, 45317, 45666, 46666, 47575, 48484, 49393, 51111, 51115, 51122, 51212, 51432, 51515, 52341, 52424, 53333, 53416, 54242, 54325, 54757, 55757, 56666, 57575, 58484, 59393, 61111, 61115, 61122, 61432, 61515, 62341, 62424, 63333, 63416, 63848, 64848, 65757, 66666, 67575, 68484, 69393, 71432, 71515, 72341, 72424, 72939, 73939, 74848, 75757, 76666, 77575, 78484, 79393, 81432, 81515, 83939, 84848, 85757, 86666, 87575, 88484, 89393, 93939, 94848, 95757, 96666, 97575, 98484, 111212, 112121, 121212, 122114, 132424, 133333, 142424, 151515, 152341, 161432, 212121, 221212, 222114, 223333, 224242, 233333, 242424, 251515, 252341, 261432, 312121, 314242, 315151, 321212, 322114, 323416, 324242, 333333, 342424, 351515, 352341, 361432, 412121, 413416, 414325, 415151, 421212, 423416, 424242, 433333, 442424, 451515, 452341, 512121, 514325, 515151, 523416, 524242, 533333, 542424, 614325, 615151, 623416, 624242, 633333, 714325, 715151, 723416, 724242, 814325, 815151, 1112121, 1121212, 1212121, 2121212, 2212121, 3121212, 3212121, 4121212, 4212121, 5121212}

Good job, thanks Jean-Marc!

à+

É.