## Friday, June 27, 2014

### A sequence of reasons

Eric Angelini recently posted to the Sequence Fanatics discussion list a very nice 1, 2, 3, 5, 11, 12, 4, 8, 7, ... which ended up as A243357: The lexicographically earliest sequence (not reusing any terms) with the property that if a vertical line is drawn between any pair of adjacent digits (commas and spaces excluded), the number Z formed by the digits to the left of the line is divisible by the final digit of Z. So, a line not just between the terms 3 and 5 (say), or the term 5 and the 1 of the following 11, but also between the two ones of that 11. To wit:

1/1 = 1
12/2 = 6
123/3 = 41
1235/5 = 247
12351/1 = 12351, 123511/1 = 123511
1235111/1 = 1235111, 12351112/2 = 6175556
123511124/4 = 30877781
1235111248/8 = 154388906
12351112487/7 = 1764444641
etc.

Not every number will appear in A243357. Nonnegative integers that are not in A243357 ended up as A244033:

0, 10, 14, 18, 20, 30, 34, 38, 40, 50, 54, 58, 60, 70, 74, 78, 80, 90, 94, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 114, 118, 120, 130, 134, 138, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 154, 158, 160, 170, 174, 178, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 194, 198, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 214, 218, 220, 228, 230, 234, 238, 240, 250, 254, 258, 260, 268, 270, 274, 278, 280, 290, 294, 298, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 313, 314, 316, 318, 319, 320, 323, 326, 329, 330, 334, 338, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 353, 354, 356, 358, 359, 360, 370, 373, 374, 376, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 394, 398, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 414, 418, 420, 428, 430, 434, 438, 440, 450, 454, 458, 460, 468, 470, 474, 478, 480, 490, 494, 498, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 514, 518, 520, 530, 534, 538, 540, 541, 542, 543, 544, 545, 546, 547, 548, 549, 550, 554, 558, 560, 570, 574, 578, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 594, 598, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 613, 614, 616, 618, 619, 620, 623, 626, 628, 629, 630, 634, 638, 640, 643, 646, 649, 650, 653, 654, 656, 658, 659, 660, 668, 670, 673, 674, 676, 678, 679, 680, 683, 686, 689, 690, 694, 698, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 714, 717, 718, 720, 727, 730, 734, 737, 738, 740, 741, 742, 743, 744, 745, 746, 747, 748, 749, 750, 754, 757, 758, 760, 767, 770, 774, 778, 780, 781, 782, 783, 784, 785, 786, 787, 788, 789, 790, 794, 797, 798, 800, 801, 802, 803, 804, 805, 806, 807, 808, 809, 810, 814, 818, 820, 828, 830, 834, 838, 840, 850, 854, 858, 860, 868, 870, 874, 878, 880, 890, 894, 898, 900, 901, 902, 903, 904, 905, 906, 907, 908, 909, 910, 913, 914, 916, 918, 919, 920, 923, 926, 929, 930, 934, 938, 939, 940, 941, 942, 943, 944, 945, 946, 947, 948, 949, 950, 953, 954, 956, 958, 959, 960, 969, 970, 973, 974, 976, 978, 979, 980, 981, 982, 983, 984, 985, 986, 987, 988, 989, 990, 994, 998, 1000, ...

Most obviously, any number containing a zero digit will have been excluded because division by zero is a no-no. Let's call this reason #1. We can exclude from A244033 those numbers that fail because of reason #1. What remains ended up as A244034:

14, 18, 34, 38, 54, 58, 74, 78, 94, 98, 114, 118, 134, 138, 141, 142, 143, 144, 145, 146, 147, 148, 149, 154, 158, 174, 178, 181, 182, 183, 184, 185, 186, 187, 188, 189, 194, 198, 214, 218, 228, 234, 238, 254, 258, 268, 274, 278, 294, 298, 313, 314, 316, 318, 319, 323, 326, 329, 334, 338, 341, 342, 343, 344, 345, 346, 347, 348, 349, 353, 354, 356, 358, 359, 373, 374, 376, 378, 379, 381, 382, 383, 384, 385, 386, 387, 388, 389, 394, 398, 414, 418, 428, 434, 438, 454, 458, 468, 474, 478, 494, 498, 514, 518, 534, 538, 541, 542, 543, 544, 545, 546, 547, 548, 549, 554, 558, 574, 578, 581, 582, 583, 584, 585, 586, 587, 588, 589, 594, 598, 613, 614, 616, 618, 619, 623, 626, 628, 629, 634, 638, 643, 646, 649, 653, 654, 656, 658, 659, 668, 673, 674, 676, 678, 679, 683, 686, 689, 694, 698, 714, 717, 718, 727, 734, 737, 738, 741, 742, 743, 744, 745, 746, 747, 748, 749, 754, 757, 758, 767, 774, 778, 781, 782, 783, 784, 785, 786, 787, 788, 789, 794, 797, 798, 814, 818, 828, 834, 838, 854, 858, 868, 874, 878, 894, 898, 913, 914, 916, 918, 919, 923, 926, 929, 934, 938, 939, 941, 942, 943, 944, 945, 946, 947, 948, 949, 953, 954, 956, 958, 959, 969, 973, 974, 976, 978, 979, 981, 982, 983, 984, 985, 986, 987, 988, 989, 994, 998, 1114, ...

Why do these numbers fail to be included in A243357? Examining the initial ten terms, one realizes soon enough that multiples of 4 ending in 4 and multiples of 8 ending in 8 will always have an even digit preceding the 4 or 8. Therefore 14, 18, 34, 38, 54, 58, 74, 78, 94, 98, or any larger number containing these as a substring, will have been excluded for that reason. Let's call this reason #2. We can exclude from A244034 those numbers that fail because of reason #2. What remains is:

228, 268, 313, 316, 319, 323, 326, 329, 353, 356, 359, 373, 376, 379, 428, 468, 613, 616, 619, 623, 626, 628, 629, 643, 646, 649, 653, 656, 659, 668, 673, 676, 679, 683, 686, 689, 717, 727, 737, 757, 767, 797, 828, 868, 913, 916, 919, 923, 926, 929, 939, 953, 956, 959, 969, 973, 976, 979, 1228, ...

Why do these numbers fail to be included in A243357? Not because of reason #1. Not because of reason #2. You can see where I'm going with this. Without further ado, I present reasons to exclude from A243357 certain numbers:

#1: numbers containing one or more of the single-digit 0.

#2: numbers containing as a substring any one or more of the 2-digit strings 14, 18, 34, 38, 54, 58, 74, 78, 94, 98.

#3: numbers containing as a substring any one or more of the 3-digit strings 228, 268, 313, 316, 319, 323, 326, 329, 353, 356, 359, 373, 376, 379, 428, 468, 613, 616, 619, 623, 626, 628, 629, 643, 646, 649, 653, 656, 659, 668, 673, 676, 679, 683, 686, 689, 717, 727, 737, 757, 767, 797, 828, 868, 913, 916, 919, 923, 926, 929, 939, 953, 956, 959, 969, 973, 976, 979.

#4: numbers containing as a substring any one or more of the 4-digit strings 3113, 3116, 3119, 3173, 3176, 3179, 3223, 3226, 3229, 3253, 3256, 3259, 3283, 3286, 3289, 3523, 3526, 3529, 3553, 3556, 3559, 3713, 3716, 3719, 3773, 3776, 3779, 6113, 6116, 6119, 6173, 6176, 6179, 6223, 6226, 6229, 6253, 6256, 6259, 6413, 6416, 6419, 6443, 6446, 6449, 6473, 6476, 6479, 6523, 6526, 6529, 6553, 6556, 6559, 6713, 6716, 6719, 6773, 6776, 6779, 6823, 6826, 6829, 6853, 6856, 6859, 6883, 6886, 6889, 7117, 7127, 7137, 7157, 7167, 7197, 7227, 7237, 7247, 7257, 7267, 7297, 7317, 7327, 7337, 7367, 7397, 7517, 7527, 7537, 7557, 7597, 7617, 7627, 7647, 7657, 7667, 7687, 7697, 7927, 7937, 7957, 7967, 7997, 9113, 9116, 9119, 9129, 9159, 9173, 9176, 9179, 9219, 9223, 9226, 9229, 9249, 9253, 9256, 9259, 9283, 9286, 9289, 9339, 9519, 9523, 9526, 9529, 9553, 9556, 9559, 9579, 9669, 9713, 9716, 9719, 9759, 9773, 9776, 9779.

#5: numbers containing as a substring any one or more of the 5-digit strings 31123, 31126, 31129, 31153, 31156, 31159, 31213, 31216, 31219, 31223, 31226, 31229, 31243, 31246, 31249, 31253, 31256, 31259, 31273, 31276, 31279, 31283, 31286, 31289, 31513, 31516, 31519, 31523, 31526, 31529, 31553, 31556, 31559, 31573, 31576, 31579, 31723, 31726, 31729, 31753, 31756, 31759, 32113, 32116, 32119, 32123, 32126, 32129, 32153, 32156, 32159, 32173, 32176, 32179, 32213, 32216, 32219, 32243, 32246, 32249, 32273, 32276, 32279, 32413, 32416, 32419, 32423, 32426, 32429, 32443, 32446, 32449, 32453, 32456, 32459, 32473, 32476, 32479, 32483, 32486, 32489, 32513, 32516, 32519, 32573, 32576, 32579, 32713, 32716, 32719, 32723, 32726, 32729, 32753, 32756, 32759, 32773, 32776, 32779, 32813, 32816, 32819, 32843, 32846, 32849, 32873, 32876, 32879, 35113, 35116, 35119, 35123, 35126, 35129, 35153, 35156, 35159, 35173, 35176, 35179, 35213, 35216, 35219, 35243, 35246, 35249, 35273, 35276, 35279, 35513, 35516, 35519, 35573, 35576, 35579, 35713, 35716, 35719, 35723, 35726, 35729, 35753, 35756, 35759, 35773, 35776, 35779, 37123, 37126, 37129, 37153, 37156, 37159, 37213, 37216, 37219, 37223, 37226, 37229, 37243, 37246, 37249, 37253, 37256, 37259, 37283, 37286, 37289, 37513, 37516, 37519, 37523, 37526, 37529, 37553, 37556, 37559, 37723, 37726, 37729, 37753, 37756, 37759, 61123, 61126, 61129, 61153, 61156, 61159, 61213, 61216, 61219, 61223, 61226, 61229, 61243, 61246, 61249, 61253, 61256, 61259, 61273, 61276, 61279, 61283, 61286, 61289, 61513, 61516, 61519, 61523, 61526, 61529, 61553, 61556, 61559, 61573, 61576, 61579, 61723, 61726, 61729, 61753, 61756, 61759, 62113, 62116, 62119, 62123, 62126, 62129, 62153, 62156, 62159, 62173, 62176, 62179, 62213, 62216, 62219, 62243, 62246, 62249, 62273, 62276, 62279, 62413, 62416, 62419, 62423, 62426, 62429, 62443, 62446, 62449, 62453, 62456, 62459, 62473, 62476, 62479, 62483, 62486, 62489, 62513, 62516, 62519, 62573, 62576, 62579, 62713, 62716, 62719, 62723, 62726, 62729, 62753, 62756, 62759, 62773, 62776, 62779, 64123, 64126, 64129, 64153, 64156, 64159, 64213, 64216, 64219, 64223, 64226, 64229, 64243, 64246, 64249, 64253, 64256, 64259, 64273, 64276, 64279, 64423, 64426, 64429, 64453, 64456, 64459, 64483, 64486, 64489, 64513, 64516, 64519, 64523, 64526, 64529, 64553, 64556, 64559, 64573, 64576, 64579, 64723, 64726, 64729, 64753, 64756, 64759, 64813, 64816, 64819, 64823, 64826, 64829, 64843, 64846, 64849, 64853, 64856, 64859, 64873, 64876, 64879, 64883, 64886, 64889, 65113, 65116, 65119, 65123, 65126, 65129, 65153, 65156, 65159, 65173, 65176, 65179, 65213, 65216, 65219, 65243, 65246, 65249, 65273, 65276, 65279, 65513, 65516, 65519, 65573, 65576, 65579, 65713, 65716, 65719, 65723, 65726, 65729, 65753, 65756, 65759, 65773, 65776, 65779, 67123, 67126, 67129, 67153, 67156, 67159, 67213, 67216, 67219, 67223, 67226, 67229, 67243, 67246, 67249, 67253, 67256, 67259, 67283, 67286, 67289, 67513, 67516, 67519, 67523, 67526, 67529, 67553, 67556, 67559, 67723, 67726, 67729, 67753, 67756, 67759, 68113, 68116, 68119, 68123, 68126, 68129, 68153, 68156, 68159, 68173, 68176, 68179, 68213, 68216, 68219, 68243, 68246, 68249, 68273, 68276, 68279, 68413, 68416, 68419, 68423, 68426, 68429, 68443, 68446, 68449, 68453, 68456, 68459, 68473, 68476, 68479, 68483, 68486, 68489, 68513, 68516, 68519, 68573, 68576, 68579, 68713, 68716, 68719, 68723, 68726, 68729, 68753, 68756, 68759, 68773, 68776, 68779, 68813, 68816, 68819, 68843, 68846, 68849, 68873, 68876, 68879, 71117, 71137, 71157, 71167, 71217, 71227, 71237, 71247, 71257, 71287, 71297, 71317, 71327, 71357, 71367, 71397, 71517, 71527, 71537, 71557, 71567, 71597, 71627, 71637, 71647, 71657, 71667, 71697, 71917, 71927, 71937, 71957, 71997, 72117, 72127, 72137, 72157, 72167, 72197, 72217, 72227, 72237, 72257, 72267, 72297, 72327, 72337, 72357, 72367, 72397, 72417, 72427, 72437, 72447, 72467, 72487, 72497, 72517, 72537, 72557, 72567, 72617, 72627, 72637, 72647, 72657, 72697, 72817, 72827, 72837, 72847, 72857, 72867, 72887, 72897, 72917, 72927, 72937, 72957, 72967, 72997, 73117, 73127, 73217, 73247, 73257, 73287, 73317, 73327, 73337, 73357, 73397, 73517, 73527, 73557, 73617, 73627, 73637, 73657, 73667, 73687, 73697, 73917, 73937, 73957, 73967, 75127, 75137, 75157, 75167, 75197, 75217, 75227, 75237, 75247, 75267, 75287, 75297, 75317, 75337, 75357, 75367, 75517, 75527, 75557, 75567, 75597, 75617, 75627, 75637, 75647, 75657, 75667, 75687, 75697, 75917, 75927, 75937, 75967, 75997, 76117, 76127, 76157, 76217, 76227, 76247, 76257, 76317, 76327, 76337, 76357, 76367, 76397, 76417, 76427, 76457, 76487, 76527, 76557, 76617, 76627, 76637, 76647, 76667, 76697, 76817, 76827, 76847, 76857, 76887, 76917, 76927, 76957, 76967, 76997, 79117, 79127, 79157, 79217, 79227, 79257, 79287, 79327, 79337, 79357, 79367, 79517, 79557, 79617, 79627, 79637, 79647, 79657, 79687, 79917, 79927, 79937, 79957, 79967, 79997, 91119, 91123, 91126, 91129, 91153, 91156, 91159, 91213, 91216, 91219, 91223, 91226, 91229, 91239, 91243, 91246, 91249, 91253, 91256, 91259, 91273, 91276, 91279, 91283, 91286, 91289, 91513, 91516, 91519, 91523, 91526, 91529, 91553, 91556, 91559, 91569, 91573, 91576, 91579, 91723, 91726, 91729, 91753, 91756, 91759, 91779, 92113, 92116, 92119, 92123, 92126, 92129, 92139, 92153, 92156, 92159, 92173, 92176, 92179, 92213, 92216, 92219, 92229, 92243, 92246, 92249, 92273, 92276, 92279, 92413, 92416, 92419, 92423, 92426, 92429, 92443, 92446, 92449, 92453, 92456, 92459, 92469, 92473, 92476, 92479, 92483, 92486, 92489, 92513, 92516, 92519, 92559, 92573, 92576, 92579, 92713, 92716, 92719, 92723, 92726, 92729, 92739, 92753, 92756, 92759, 92769, 92773, 92776, 92779, 92813, 92816, 92819, 92829, 92843, 92846, 92849, 92859, 92873, 92876, 92879, 93129, 93219, 93279, 93369, 93579, 93639, 93669, 93729, 93759, 95113, 95116, 95119, 95123, 95126, 95129, 95153, 95156, 95159, 95169, 95173, 95176, 95179, 95213, 95216, 95219, 95243, 95246, 95249, 95259, 95273, 95276, 95279, 95289, 95513, 95516, 95519, 95529, 95559, 95573, 95576, 95579, 95713, 95716, 95719, 95723, 95726, 95729, 95739, 95753, 95756, 95759, 95773, 95776, 95779, 96159, 96249, 96279, 96339, 96369, 96429, 96459, 96519, 96639, 96729, 96819, 96879, 97123, 97126, 97129, 97153, 97156, 97159, 97213, 97216, 97219, 97223, 97226, 97229, 97239, 97243, 97246, 97249, 97253, 97256, 97259, 97269, 97283, 97286, 97289, 97513, 97516, 97519, 97523, 97526, 97529, 97539, 97553, 97556, 97559, 97719, 97723, 97726, 97729, 97753, 97756, 97759, 97779.

#6: numbers containing as a substring any one or more of the 6-digit strings 311123, 311126, 311129, 311153, 311156, 311159, 311213, 311216, 311219, 311243, 311246, 311249, 311273, 311276, 311279, 311513, 311516, 311519, 311573, 311576, 311579, 311723, 311726, 311729, 311753, 311756, 311759, 312113, 312116, 312119, 312173, 312176, 312179, 312413, 312416, 312419, 312443, 312446, 312449, 312473, 312476, 312479, 312713, 312716, 312719, 312773, 312776, 312779, 315113, 315116, 315119, 315173, 315176, 315179, 315713, 315716, 315719, 315773, 315776, 315779, 317123, 317126, 317129, 317153, 317156, 317159, 317213, 317216, 317219, 317243, 317246, 317249, 317513, 317516, 317519, 317723, 317726, 317729, 317753, 317756, 317759, 321113, 321116, 321119, 321173, 321176, 321179, 321713, 321716, 321719, 321773, 321776, 321779, 322223, 322226, 322229, 322253, 322256, 322259, 322523, 322526, 322529, 322553, 322556, 322559, 324113, 324116, 324119, 324173, 324176, 324179, 324413, 324416, 324419, 324443, 324446, 324449, 324473, 324476, 324479, 324713, 324716, 324719, 324773, 324776, 324779, 325223, 325226, 325229, 325253, 325256, 325259, 325283, 325286, 325289, 325523, 325526, 325529, 325553, 325556, 325559, 327113, 327116, 327119, 327713, 327716, 327719, 327773, 327776, 327779, 328223, 328226, 328229, 328253, 328256, 328259, 328523, 328526, 328529, 328553, 328556, 328559, 328823, 328826, 328829, 328853, 328856, 328859, 328883, 328886, 328889, 351113, 351116, 351119, 351173, 351176, 351179, 351713, 351716, 351719, 351773, 351776, 351779, 352223, 352226, 352229, 352253, 352256, 352259, 352523, 352526, 352529, 352553, 352556, 352559, 352823, 352826, 352829, 352853, 352856, 352859, 352883, 352886, 352889, 355223, 355226, 355229, 355253, 355256, 355259, 355283, 355286, 355289, 355523, 355526, 355529, 355553, 355556, 355559, 357113, 357116, 357119, 357713, 357716, 357719, 357773, 357776, 357779, 371123, 371126, 371129, 371153, 371156, 371159, 371213, 371216, 371219, 371243, 371246, 371249, 371513, 371516, 371519, 372113, 372116, 372119, 372173, 372176, 372179, 372413, 372416, 372419, 372443, 372446, 372449, 375113, 375116, 375119, 377123, 377126, 377129, 377153, 377156, 377159, 377213, 377216, 377219, 377243, 377246, 377249, 377513, 377516, 377519, 377723, 377726, 377729, 377753, 377756, 377759, 611123, 611126, 611129, 611153, 611156, 611159, 611213, 611216, 611219, 611243, 611246, 611249, 611273, 611276, 611279, 611513, 611516, 611519, 611573, 611576, 611579, 611723, 611726, 611729, 611753, 611756, 611759, 612113, 612116, 612119, 612173, 612176, 612179, 612413, 612416, 612419, 612443, 612446, 612449, 612473, 612476, 612479, 612713, 612716, 612719, 612773, 612776, 612779, 615113, 615116, 615119, 615173, 615176, 615179, 615713, 615716, 615719, 615773, 615776, 615779, 617123, 617126, 617129, 617153, 617156, 617159, 617213, 617216, 617219, 617243, 617246, 617249, 617513, 617516, 617519, 617723, 617726, 617729, 617753, 617756, 617759, 621113, 621116, 621119, 621173, 621176, 621179, 621713, 621716, 621719, 621773, 621776, 621779, 622223, 622226, 622229, 622253, 622256, 622259, 622523, 622526, 622529, 622553, 622556, 622559, 624113, 624116, 624119, 624173, 624176, 624179, 624413, 624416, 624419, 624443, 624446, 624449, 624473, 624476, 624479, 624713, 624716, 624719, 624773, 624776, 624779, 625223, 625226, 625229, 625253, 625256, 625259, 625283, 625286, 625289, 625523, 625526, 625529, 625553, 625556, 625559, 627113, 627116, 627119, 627713, 627716, 627719, 627773, 627776, 627779, 641123, 641126, 641129, 641153, 641156, 641159, 641213, 641216, 641219, 641243, 641246, 641249, 641273, 641276, 641279, 641513, 641516, 641519, 641573, 641576, 641579, 641723, 641726, 641729, 641753, 641756, 641759, 642113, 642116, 642119, 642173, 642176, 642179, 642413, 642416, 642419, 642443, 642446, 642449, 642473, 642476, 642479, 642713, 642716, 642719, 642773, 642776, 642779, 644123, 644126, 644129, 644153, 644156, 644159, 644213, 644216, 644219, 644243, 644246, 644249, 644273, 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792487, 792517, 792527, 792557, 792847, 792857, 793117, 793127, 793157, 793217, 793227, 793257, 793287, 793327, 793337, 793357, 793367, 793557, 793617, 793627, 793637, 793647, 793687, 793697, 795117, 795157, 795167, 795217, 795227, 795247, 795257, 795287, 795517, 795527, 796127, 796157, 796217, 796227, 796337, 796357, 796367, 796417, 796427, 796447, 796487, 796517, 796557, 796617, 796627, 796637, 796647, 796657, 796827, 796847, 796857, 799157, 799217, 799227, 799247, 799287, 799317, 799357, 799367, 799517, 799527, 799557, 799627, 799637, 799647, 799657, 799667, 799917, 799927, 799937, 799997, 911123, 911126, 911129, 911139, 911153, 911156, 911159, 911179, 911213, 911216, 911219, 911229, 911243, 911246, 911249, 911273, 911276, 911279, 911289, 911513, 911516, 911519, 911559, 911573, 911576, 911579, 911719, 911723, 911726, 911729, 911739, 911753, 911756, 911759, 911769, 911779, 912113, 912116, 912119, 912129, 912173, 912176, 912179, 912219, 912229, 912259, 912279, 912369, 912413, 912416, 912419, 912443, 912446, 912449, 912459, 912473, 912476, 912479, 912489, 912529, 912579, 912639, 912669, 912713, 912716, 912719, 912729, 912759, 912773, 912776, 912779, 912819, 912849, 912859, 912889, 915113, 915116, 915119, 915159, 915173, 915176, 915179, 915229, 915249, 915279, 915289, 915339, 915369, 915519, 915559, 915639, 915713, 915716, 915719, 915729, 915773, 915776, 915779, 917119, 917123, 917126, 917129, 917139, 917153, 917156, 917159, 917169, 917213, 917216, 917219, 917229, 917243, 917246, 917249, 917259, 917513, 917516, 917519, 917529, 917719, 917723, 917726, 917729, 917753, 917756, 917759, 917769, 921113, 921116, 921119, 921129, 921173, 921176, 921179, 921219, 921229, 921259, 921279, 921369, 921529, 921579, 921639, 921669, 921713, 921716, 921719, 921729, 921759, 921773, 921776, 921779, 922119, 922129, 922159, 922179, 922219, 922223, 922226, 922229, 922249, 922253, 922256, 922259, 922269, 922429, 922449, 922479, 922489, 922519, 922523, 922526, 922529, 922539, 922553, 922556, 922559, 922569, 922579, 922719, 922759, 924113, 924116, 924119, 924159, 924173, 924176, 924179, 924229, 924249, 924279, 924339, 924369, 924413, 924416, 924419, 924429, 924443, 924446, 924449, 924459, 924473, 924476, 924479, 924519, 924559, 924639, 924713, 924716, 924719, 924729, 924773, 924776, 924779, 924819, 924829, 924859, 924879, 925129, 925179, 925219, 925223, 925226, 925229, 925239, 925253, 925256, 925259, 925269, 925279, 925283, 925286, 925289, 925523, 925526, 925529, 925539, 925553, 925556, 925559, 925579, 925719, 925729, 925759, 925779, 927113, 927116, 927119, 927129, 927159, 927219, 927249, 927259, 927289, 927339, 927519, 927529, 927559, 927669, 927713, 927716, 927719, 927759, 927773, 927776, 927779, 928119, 928159, 928223, 928226, 928229, 928239, 928249, 928253, 928256, 928259, 928279, 928419, 928429, 928459, 928479, 928519, 928523, 928526, 928529, 928553, 928556, 928559, 928569, 928729, 928779, 928819, 928823, 928826, 928829, 928839, 928853, 928856, 928859, 928869, 928879, 928883, 928886, 928889, 931119, 931179, 931269, 931539, 931569, 931719, 932169, 932259, 932439, 932469, 932529, 932559, 932739, 932829, 932889, 933159, 933249, 933279, 933339, 933369, 933519, 933639, 933729, 935139, 935169, 935229, 935259, 935529, 935769, 936129, 936159, 936219, 936249, 936339, 936429, 936489, 936519, 936579, 936669, 936759, 936849, 936879, 937119, 937239, 937569, 937779, 951113, 951116, 951119, 951159, 951173, 951176, 951179, 951229, 951249, 951279, 951289, 951339, 951369, 951519, 951559, 951639, 951713, 951716, 951719, 951729, 951773, 951776, 951779, 952129, 952179, 952219, 952223, 952226, 952229, 952239, 952253, 952256, 952259, 952269, 952279, 952419, 952449, 952459, 952489, 952523, 952526, 952529, 952539, 952553, 952556, 952559, 952579, 952719, 952729, 952759, 952779, 952819, 952823, 952826, 952829, 952849, 952853, 952856, 952859, 952869, 952883, 952886, 952889, 955119, 955159, 955223, 955226, 955229, 955239, 955249, 955253, 955256, 955259, 955279, 955283, 955286, 955289, 955519, 955523, 955526, 955529, 955553, 955556, 955559, 955569, 955729, 955779, 957113, 957116, 957119, 957129, 957219, 957229, 957259, 957369, 957529, 957639, 957669, 957713, 957716, 957719, 957729, 957759, 957773, 957776, 957779, 961179, 961239, 961269, 961539, 961719, 961779, 962139, 962169, 962229, 962259, 962439, 962529, 962769, 963129, 963159, 963219, 963249, 963339, 963519, 963579, 963669, 963759, 964119, 964239, 964419, 964479, 964569, 964779, 964839, 964869, 965139, 965229, 965289, 965559, 965739, 965769, 966129, 966219, 966279, 966369, 966459, 966489, 966579, 966639, 966669, 966729, 966759, 967119, 967269, 967539, 967569, 967719, 968169, 968259, 968439, 968469, 968529, 968559, 968739, 968829, 968889, 971119, 971123, 971126, 971129, 971139, 971153, 971156, 971159, 971169, 971213, 971216, 971219, 971229, 971243, 971246, 971249, 971259, 971513, 971516, 971519, 971529, 972113, 972116, 972119, 972129, 972159, 972173, 972176, 972179, 972219, 972249, 972259, 972339, 972413, 972416, 972419, 972429, 972443, 972446, 972449, 972489, 972519, 972529, 972559, 972669, 972829, 972849, 972879, 972889, 975113, 975116, 975119, 975129, 975219, 975229, 975259, 975369, 975529, 975639, 975669, 977119, 977123, 977126, 977129, 977153, 977156, 977159, 977169, 977213, 977216, 977219, 977243, 977246, 977249, 977259, 977289, 977513, 977516, 977519, 977529, 977559, 977723, 977726, 977729, 977739, 977753, 977756, 977759, 977779.

And so on. Obviously, numbers may fail to be included in A243357 for more than one reason. For example, 140 fails because of reason #1 and reason #2. The numbers in each reason are a kind of puzzle asking you to determine why that number fails. Once the eight integers ending with 8 in reason #3 are dispensed with, it appears that every other number involves some sort of division complication of integers beginning and ending with 3, 6, or 9, or beginning and ending with 7.