Friday, June 27

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, 644276, 644279, 644423, 644426, 644429, 644453, 644456, 644459, 644483, 644486, 644489, 644513, 644516, 644519, 644573, 644576, 644579, 644723, 644726, 644729, 644753, 644756, 644759, 644813, 644816, 644819, 644843, 644846, 644849, 644873, 644876, 644879, 645113, 645116, 645119, 645173, 645176, 645179, 645713, 645716, 645719, 645773, 645776, 645779, 647123, 647126, 647129, 647153, 647156, 647159, 647213, 647216, 647219, 647243, 647246, 647249, 647513, 647516, 647519, 647723, 647726, 647729, 647753, 647756, 647759, 648113, 648116, 648119, 648173, 648176, 648179, 648413, 648416, 648419, 648443, 648446, 648449, 648473, 648476, 648479, 648713, 648716, 648719, 648773, 648776, 648779, 651113, 651116, 651119, 651173, 651176, 651179, 651713, 651716, 651719, 651773, 651776, 651779, 652223, 652226, 652229, 652253, 652256, 652259, 652523, 652526, 652529, 652553, 652556, 652559, 652823, 652826, 652829, 652853, 652856, 652859, 652883, 652886, 652889, 655223, 655226, 655229, 655253, 655256, 655259, 655283, 655286, 655289, 655523, 655526, 655529, 655553, 655556, 655559, 657113, 657116, 657119, 657713, 657716, 657719, 657773, 657776, 657779, 671123, 671126, 671129, 671153, 671156, 671159, 671213, 671216, 671219, 671243, 671246, 671249, 671513, 671516, 671519, 672113, 672116, 672119, 672173, 672176, 672179, 672413, 672416, 672419, 672443, 672446, 672449, 675113, 675116, 675119, 677123, 677126, 677129, 677153, 677156, 677159, 677213, 677216, 677219, 677243, 677246, 677249, 677513, 677516, 677519, 677723, 677726, 677729, 677753, 677756, 677759, 681113, 681116, 681119, 681173, 681176, 681179, 681713, 681716, 681719, 681773, 681776, 681779, 682223, 682226, 682229, 682253, 682256, 682259, 682523, 682526, 682529, 682553, 682556, 682559, 684113, 684116, 684119, 684173, 684176, 684179, 684413, 684416, 684419, 684443, 684446, 684449, 684473, 684476, 684479, 684713, 684716, 684719, 684773, 684776, 684779, 685223, 685226, 685229, 685253, 685256, 685259, 685283, 685286, 685289, 685523, 685526, 685529, 685553, 685556, 685559, 687113, 687116, 687119, 687713, 687716, 687719, 687773, 687776, 687779, 688223, 688226, 688229, 688253, 688256, 688259, 688523, 688526, 688529, 688553, 688556, 688559, 688823, 688826, 688829, 688853, 688856, 688859, 688883, 688886, 688889, 711117, 711157, 711167, 711217, 711227, 711237, 711247, 711257, 711287, 711297, 711317, 711327, 711357, 711367, 711397, 711517, 711527, 711537, 711567, 711597, 711637, 711647, 711657, 711667, 711917, 711927, 711937, 711957, 711997, 712127, 712137, 712157, 712167, 712197, 712217, 712227, 712237, 712267, 712297, 712337, 712357, 712367, 712417, 712427, 712437, 712447, 712487, 712497, 712517, 712557, 712567, 712617, 712627, 712637, 712647, 712657, 712697, 712827, 712837, 712847, 712857, 712867, 712897, 712917, 712927, 712937, 712967, 712997, 713117, 713127, 713217, 713247, 713257, 713287, 713317, 713327, 713337, 713357, 713397, 713527, 713557, 713617, 713627, 713637, 713667, 713687, 713697, 713917, 713957, 713967, 715137, 715157, 715167, 715217, 715227, 715237, 715247, 715287, 715297, 715317, 715357, 715367, 715517, 715527, 715557, 715567, 715597, 715627, 715637, 715647, 715657, 715667, 715697, 715917, 715927, 715937, 715997, 716117, 716127, 716157, 716217, 716227, 716257, 716327, 716337, 716357, 716367, 716397, 716417, 716427, 716487, 716557, 716617, 716627, 716637, 716647, 716697, 716817, 716827, 716847, 716857, 716887, 716917, 716927, 716957, 716967, 716997, 719127, 719157, 719217, 719227, 719287, 719337, 719357, 719367, 719517, 719557, 719617, 719627, 719637, 719647, 719657, 719687, 719917, 719927, 719937, 719967, 719997, 721117, 721127, 721157, 721167, 721197, 721227, 721237, 721247, 721257, 721267, 721297, 721317, 721327, 721337, 721367, 721397, 721517, 721527, 721537, 721597, 721617, 721647, 721657, 721667, 721687, 721927, 721937, 721957, 721967, 721997, 722137, 722157, 722167, 722217, 722227, 722237, 722247, 722297, 722317, 722357, 722367, 722417, 722427, 722437, 722447, 722457, 722487, 722497, 722517, 722527, 722557, 722567, 722597, 722627, 722637, 722647, 722657, 722667, 722697, 722917, 722927, 722937, 722997, 723117, 723127, 723157, 723217, 723227, 723257, 723287, 723327, 723337, 723357, 723367, 723397, 723557, 723617, 723627, 723637, 723647, 723687, 723697, 723917, 723927, 723957, 723967, 723997, 724117, 724127, 724137, 724167, 724197, 724237, 724247, 724257, 724267, 724317, 724327, 724337, 724397, 724417, 724447, 724457, 724467, 724487, 724517, 724527, 724537, 724557, 724597, 724617, 724627, 724657, 724667, 724697, 724817, 724827, 724837, 724867, 724887, 724897, 724937, 724957, 724967, 725117, 725157, 725167, 725217, 725227, 725237, 725247, 725257, 725287, 725297, 725317, 725327, 725357, 725367, 725397, 725517, 725527, 725537, 725567, 725597, 725637, 725647, 725657, 725667, 725917, 725927, 725937, 725957, 725997, 726127, 726157, 726217, 726227, 726337, 726357, 726367, 726417, 726427, 726447, 726487, 726517, 726557, 726617, 726627, 726637, 726647, 726657, 726697, 726917, 726927, 726937, 726967, 726997, 728117, 728127, 728157, 728167, 728197, 728227, 728237, 728247, 728257, 728267, 728297, 728317, 728327, 728337, 728367, 728397, 728437, 728447, 728457, 728467, 728517, 728527, 728537, 728597, 728617, 728647, 728657, 728667, 728817, 728827, 728857, 728867, 728887, 728897, 728927, 728937, 728957, 728967, 728997, 729157, 729217, 729227, 729247, 729287, 729317, 729357, 729367, 729517, 729527, 729557, 729627, 729637, 729647, 729657, 729667, 729917, 729927, 729937, 729997, 731117, 731127, 731237, 731247, 731257, 731267, 731517, 731527, 731537, 731557, 731597, 732117, 732157, 732167, 732217, 732227, 732247, 732257, 732427, 732437, 732447, 732457, 732467, 732497, 732517, 732527, 732817, 732847, 732857, 732887, 733127, 733157, 733217, 733227, 733287, 733337, 733357, 733367, 733517, 733557, 733617, 733627, 733637, 733647, 733657, 733687, 733697, 733917, 733927, 733937, 733967, 733997, 735117, 735127, 735157, 735167, 735197, 735227, 735247, 735257, 735517, 735527, 736157, 736217, 736227, 736247, 736317, 736357, 736367, 736417, 736427, 736447, 736457, 736487, 736517, 736527, 736557, 736627, 736637, 736647, 736657, 736667, 736697, 736847, 736857, 736917, 736927, 736937, 736997, 739117, 739157, 739217, 739227, 739247, 739257, 739287, 739317, 739327, 739357, 739367, 739517, 739527, 739637, 739647, 739657, 739667, 739917, 739927, 739937, 739957, 739997, 751117, 751127, 751137, 751157, 751197, 751217, 751227, 751257, 751267, 751287, 751297, 751327, 751337, 751357, 751367, 751397, 751537, 751557, 751567, 751617, 751627, 751637, 751647, 751687, 751697, 751917, 751927, 751957, 751967, 751997, 752117, 752127, 752137, 752167, 752197, 752237, 752247, 752257, 752267, 752317, 752327, 752337, 752397, 752417, 752447, 752457, 752467, 752487, 752517, 752527, 752537, 752557, 752597, 752617, 752627, 752657, 752667, 752697, 752817, 752827, 752837, 752867, 752887, 752897, 752937, 752957, 752967, 753117, 753157, 753217, 753227, 753247, 753257, 753287, 753317, 753327, 753357, 753367, 753397, 753517, 753527, 753637, 753647, 753657, 753667, 753917, 753927, 753937, 753957, 753997, 755117, 755127, 755137, 755197, 755217, 755247, 755257, 755267, 755287, 755317, 755327, 755337, 755357, 755397, 755527, 755537, 755557, 755567, 755597, 755617, 755627, 755637, 755667, 755687, 755697, 755917, 755957, 755967, 756117, 756127, 756157, 756227, 756247, 756257, 756317, 756327, 756337, 756367, 756397, 756447, 756457, 756517, 756527, 756617, 756647, 756657, 756667, 756817, 756827, 756857, 756887, 756927, 756937, 756957, 756967, 756997, 759117, 759127, 759247, 759257, 759317, 759327, 759337, 759517, 759527, 759557, 759617, 759627, 759657, 759667, 759687, 759937, 759957, 759967, 761127, 761157, 761217, 761227, 761237, 761267, 761287, 761297, 761517, 761557, 761567, 762117, 762127, 762137, 762197, 762217, 762247, 762257, 762417, 762427, 762457, 762467, 762487, 762497, 762527, 762557, 763117, 763127, 763157, 763227, 763247, 763257, 763317, 763327, 763337, 763367, 763397, 763517, 763527, 763617, 763647, 763657, 763667, 763687, 763927, 763937, 763957, 763967, 763997, 764157, 764217, 764227, 764237, 764247, 764297, 764417, 764427, 764447, 764457, 764487, 764517, 764527, 764557, 764567, 764597, 764837, 764847, 764857, 764867, 765117, 765127, 765137, 765157, 765197, 765217, 765227, 765257, 765287, 765557, 766117, 766127, 766247, 766257, 766317, 766327, 766337, 766397, 766417, 766447, 766457, 766487, 766517, 766527, 766557, 766617, 766627, 766657, 766667, 766697, 766937, 766957, 766967, 768127, 768137, 768157, 768167, 768197, 768217, 768227, 768417, 768427, 768437, 768447, 768487, 768497, 768517, 768557, 768827, 768847, 768857, 769117, 769127, 769217, 769247, 769257, 769287, 769317, 769327, 769337, 769357, 769527, 769557, 769617, 769627, 769637, 769667, 769687, 769917, 769957, 769967, 791117, 791127, 791157, 791227, 791237, 791247, 791257, 791267, 791517, 791527, 791537, 792137, 792157, 792167, 792217, 792227, 792247, 792417, 792427, 792437, 792447, 792457, 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.

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