Generalization regarding base-b proper fractions n/d having nontrivial anomalous cancellation with 2 <= b <= 120 and d <= b^2 + b. Michael De Vlieger, St. Louis, Missouri 201709132030, updated 201709150800. A trivial anomalous cancellation involves digit k = 0 for numerator n and denominator d both such that they are congruent to 0 (mod b). In this document, the word "qualified" means that the term is in A292288 or A292289, that is, the number in question pertains to a base-b proper fraction with nontrivial anomalous cancellation. There are several parts to this document: 1. General analysis 2. Record transform and indices for i = number of qualified denominators d < b^2 3. Irregular triangle: list of qualified denominators <= b^2 + b 4. Irregular triangle: list of numerators pertaining to qualified denominators 5. Algorithm //////////// 1. General analysis. b = base and index n = A292288(b) = smallest numerator that pertains to d d = A292289(b) = smallest qualified denominator n/d = simplified ratio of numerator n and denominator d i = number of qualified denominators d < b^2 k = base-b digit cancelled in the numerator and denominator to arrive at n/d b-n+1 = difference between base and numerator plus one. b^2-d = difference between the square of the base and denominator. N = numerator pertaining to D D = largest qualified denominator D < b^2, or "-" if none. N/D = simplified ratio of numerator N and denominator D b^2-D = difference between b^2 and D b n d n/d i k b-n+1 b^2-d N D N/D b^2-D b --------------------------------------------------------------------------------- 2 3 6 1/2 0 1 0 -2 - - - - 2 3 4 12 1/3 0 1 0 -3 - - - - 3 4 7 14 1/2 1 3 2 2 7 14 1/2 2 4 5 6 30 1/5 0 1 0 -5 - - - - 5 6 11 33 1/3 2 5 4 3 17 34 1/2 2 6 7 8 56 1/7 0 1 0 -7 - - - - 7 8 15 60 1/4 2 7 6 4 31 62 1/2 2 8 9 13 39 1/3 2 4 3 42 26 78 1/3 3 9 10 16 64 1/4 4 6 5 36 49 98 1/2 2 10 11 12 132 1/11 0 1 0 -11 - - - - 11 12 23 138 1/6 4 11 10 6 71 142 1/2 2 12 13 14 182 1/13 0 1 0 -13 - - - - 13 14 27 189 1/7 2 13 12 7 97 194 1/2 2 14 15 22 110 1/5 6 7 6 115 74 222 1/3 3 15 16 21 84 1/4 7 5 4 172 127 254 1/2 2 16 17 18 306 1/17 0 1 0 -17 - - - - 17 18 35 315 1/9 4 17 16 9 161 322 1/2 2 18 19 20 380 1/19 0 1 0 -19 - - - - 19 20 39 390 1/10 4 19 18 10 199 398 1/2 2 20 21 29 174 1/6 10 8 7 267 146 438 1/3 3 21 22 34 272 1/8 6 12 11 212 241 482 1/2 2 22 23 24 552 1/23 0 1 0 -23 - - - - 23 24 47 564 1/12 6 23 22 12 287 574 1/2 2 24 25 31 155 1/5 6 6 5 470 124 620 1/5 5 25 26 67 402 1/6 4 15 40 274 337 674 1/2 2 26 27 40 360 1/9 6 13 12 369 242 726 1/3 3 27 28 37 259 1/7 10 9 8 525 391 782 1/2 2 28 29 30 870 1/29 0 1 0 -29 - - - - 29 30 59 885 1/15 6 29 28 15 449 898 1/2 2 30 31 32 992 1/31 0 1 0 -31 - - - - 31 32 63 1008 1/16 4 31 30 16 511 1022 1/2 2 32 33 45 405 1/9 8 12 11 684 362 1086 1/3 3 33 34 52 624 1/12 6 18 17 532 577 1154 1/2 2 34 35 87 609 1/7 6 17 51 616 244 1220 1/5 5 35 36 43 258 1/6 21 7 6 1038 647 1294 1/2 2 36 37 38 1406 1/37 0 1 0 -37 - - - - 37 38 75 1425 1/19 2 37 36 19 721 1442 1/2 2 38 39 58 754 1/13 6 19 18 767 506 1518 1/3 3 39 40 53 530 1/10 18 13 12 1070 799 1598 1/2 2 40 41 42 1722 1/41 0 1 0 -41 - - - - 41 42 83 1743 1/21 6 41 40 21 881 1762 1/2 2 42 43 44 1892 1/43 0 1 0 -43 - - - - 43 44 87 1914 1/22 4 43 42 22 967 1934 1/2 2 44 45 56 504 1/9 18 11 10 1521 674 2022 1/3 3 45 46 70 1120 1/16 10 24 23 996 1057 2114 1/2 2 46 47 48 2256 1/47 0 1 0 -47 - - - - 47 48 95 2280 1/24 8 47 46 24 1151 2302 1/2 2 48 49 57 399 1/7 10 8 7 2002 342 2394 1/7 7 49 50 71 1065 1/15 10 21 20 1435 1249 2498 1/2 2 50 51 122 1037 2/17 12 20 70 1564 866 2598 1/3 3 51 52 69 897 1/13 12 17 16 1807 1351 2702 1/2 2 52 53 54 2862 1/53 0 1 0 -53 - - - - 53 54 107 2889 1/27 6 53 52 27 1457 2914 1/2 2 54 55 67 670 1/10 16 12 11 2355 604 3020 1/5 5 55 56 71 852 1/12 22 15 14 2284 1567 3134 1/2 2 56 57 77 1155 1/15 14 20 19 2094 1082 3246 1/3 3 57 58 88 1760 1/20 6 30 29 1604 1681 3362 1/2 2 58 59 60 3540 1/59 0 1 0 -59 - - - - 59 60 119 3570 1/30 10 59 58 30 1799 3598 1/2 2 60 61 62 3782 1/61 0 1 0 -61 - - - - 61 62 123 3813 1/31 2 61 60 31 1921 3842 1/2 2 62 63 94 1974 1/21 12 31 30 1995 1322 3966 1/3 3 63 64 73 584 1/8 21 9 8 3512 2047 4094 1/2 2 64 65 81 1053 1/13 12 16 15 3172 844 4220 1/5 5 65 66 79 869 1/11 20 13 12 3487 2177 4354 1/2 2 66 67 68 4556 1/67 0 1 0 -67 - - - - 67 68 135 4590 1/34 4 67 66 34 2311 4622 1/2 2 68 69 93 1674 1/18 10 24 23 3087 1586 4758 1/3 3 69 70 163 1630 1/10 22 23 92 3270 2449 4898 1/2 2 70 71 72 5112 1/71 0 1 0 -71 - - - - 71 72 143 5148 1/36 10 71 70 36 2591 5182 1/2 2 72 73 74 5402 1/73 0 1 0 -73 - - - - 73 74 147 5439 1/37 2 73 72 37 2737 5474 1/2 2 74 75 112 2800 1/25 12 37 36 2825 1874 5622 1/3 3 75 76 96 1536 1/16 20 20 19 4240 2887 5774 1/2 2 76 77 178 1869 2/21 14 24 100 4060 846 5922 1/7 7 77 78 92 1104 1/12 24 14 13 4980 3041 6082 1/2 2 78 79 80 6320 1/79 0 1 0 -79 - - - - 79 80 159 6360 1/40 8 79 78 40 3199 6398 1/2 2 80 81 91 819 1/9 24 10 9 5742 2186 6558 1/3 3 81 82 124 3472 1/28 8 42 41 3252 3361 6722 1/2 2 82 83 84 6972 1/83 0 1 0 -83 - - - - 83 84 167 7014 1/42 10 83 82 42 3527 7054 1/2 2 84 85 103 1545 1/15 28 18 17 5680 1444 7220 1/5 5 85 86 217 3906 1/18 6 45 130 3490 3697 7394 1/2 2 86 87 130 3770 1/29 6 43 42 3799 2522 7566 1/3 3 87 88 117 2574 1/22 18 29 28 5170 3871 7742 1/2 2 88 89 90 8010 1/89 0 1 0 -89 - - - - 89 90 179 8055 1/45 10 89 88 45 4049 8098 1/2 2 90 91 106 1378 1/13 28 15 14 6903 1182 8274 1/7 7 91 92 210 2415 2/23 16 26 117 6049 4231 8462 1/2 2 92 93 125 3000 1/24 10 32 31 5649 2882 8646 1/3 3 93 94 142 4544 1/32 6 48 47 4292 4417 8834 1/2 2 94 95 237 4503 1/19 6 47 141 4522 1804 9020 1/5 5 95 96 115 1840 1/16 30 19 18 7376 4607 9214 1/2 2 96 97 98 9506 1/97 0 1 0 -97 - - - - 97 98 195 9555 1/49 4 97 96 49 4801 9602 1/2 2 98 99 127 2794 1/22 24 28 27 7007 3266 9798 1/3 3 99 100 111 1110 1/10 37 11 10 8890 4999 9998 1/2 2 100 101 102 10302 1/101 0 1 0 -101 - - - - 101 102 203 10353 1/51 6 101 100 51 5201 10402 1/2 2 102 103 104 10712 1/103 0 1 0 -103 - - - - 103 104 207 10764 1/52 6 103 102 52 5407 10814 1/2 2 104 105 121 1694 1/14 46 16 15 9331 3674 11022 1/3 3 105 106 160 5760 1/36 14 54 53 5476 5617 11234 1/2 2 106 107 108 11556 1/107 0 1 0 -107 - - - - 107 108 215 11610 1/54 10 107 106 54 5831 11662 1/2 2 108 109 110 11990 1/109 0 1 0 -109 - - - - 109 110 219 12045 1/55 6 109 108 55 6049 12098 1/2 2 110 111 373 4476 1/12 18 40 261 7845 4106 12318 1/3 3 111 112 149 4172 1/28 24 37 36 8372 6271 12542 1/2 2 112 113 114 12882 1/113 0 1 0 -113 - - - - 113 114 227 12939 1/57 6 113 112 57 6497 12994 1/2 2 114 115 139 2780 1/20 18 24 23 10445 2644 13220 1/5 5 115 116 146 3504 1/24 14 30 29 9952 6727 13454 1/2 2 116 117 263 3419 1/13 20 29 145 10270 4562 13686 1/3 3 117 118 178 7120 1/40 10 60 59 6804 6961 13922 1/2 2 118 119 416 7072 1/17 6 59 296 7089 2022 14154 1/7 7 119 120 137 2055 1/15 50 17 16 12345 7199 14398 1/2 2 120 Observations: 1. For prime base p, (p + 1)/(p^2 + p) simplifies to 1/p by cancelling digit k = 1 in the numerator and denominator. 2. Smallest base b for which n/d, simplified, has a numerator greater than 1 is 51. The next terms are 77 and 92. 3. Let q = the least prime divisor of composite b. The fraction with the largest qualified denominator, N/D, simplifies to 1/q, and the denominator D = b - q. ///////////// 2. Record transform for i = number of qualified denominators d < b^2. index i b ------------- 1 0 2 2 1 4 3 2 6 4 4 10 5 6 15 6 7 16 7 10 21 8 21 36 9 22 56 10 24 78 11 28 85 12 30 96 13 37 100 14 46 105 15 50 120 ///////////// 3. Irregular number triangle: list of qualified denominators less than or equal to b^2 + b for 2 <= b <= 120: 2: {6} 3: {12} 4: {14, 20} 5: {30} 6: {33, 34, 42} 7: {56} 8: {60, 62, 72} 9: {39, 78, 90} 10: {64, 65, 95, 98, 110} 11: {132} 12: {138, 140, 141, 142, 156} 13: {182} 14: {189, 194, 210} 15: {110, 111, 189, 190, 220, 222, 240} 16: {84, 150, 152, 168, 248, 252, 254, 272} 17: {306} 18: {315, 318, 321, 322, 342} 19: {380} 20: {390, 395, 396, 398, 420} 21: {174, 175, 217, 219, 324, 329, 348, 350, 434, 438, 462} 22: {272, 275, 319, 320, 473, 482, 506} 23: {552} 24: {564, 568, 570, 572, 573, 574, 600} 25: {155, 310, 410, 415, 465, 620, 650} 26: {402, 403, 663, 674, 702} 27: {360, 363, 555, 558, 720, 726, 756} 28: {259, 260, 430, 434, 518, 520, 770, 777, 780, 782, 812} 29: {870} 30: {885, 890, 894, 895, 897, 898, 930} 31: {992} 32: {1008, 1016, 1020, 1022, 1056} 33: {405, 407, 539, 543, 810, 814, 1078, 1086, 1122} 34: {624, 629, 765, 768, 1139, 1154, 1190} 35: {609, 610, 860, 861, 1218, 1220, 1260} 36: {258, 368, 369, 516, 736, 738, 774, 915, 924, 1024, 1032, 1035, 1104, 1107, 1278, 1284, 1287, 1290, 1292, 1293, 1294, 1332} 37: {1406} 38: {1425, 1442, 1482} 39: {754, 759, 1113, 1118, 1508, 1518, 1560} 40: {530, 532, 735, 736, 854, 860, 1056, 1060, 1064, 1065, 1470, 1472, 1580, 1590, 1592, 1595, 1596, 1598, 1640} 41: {1722} 42: {1743, 1750, 1757, 1758, 1761, 1762, 1806} 43: {1892} 44: {1914, 1925, 1932, 1934, 1980} 45: {504, 505, 732, 735, 1005, 1008, 1010, 1011, 1375, 1377, 1464, 1470, 1512, 1515, 2010, 2016, 2020, 2022, 2070} 46: {1120, 1127, 1170, 1173, 1265, 1266, 1403, 1408, 2093, 2114, 2162} 47: {2256} 48: {2280, 2288, 2292, 2296, 2298, 2300, 2301, 2302, 2352} 49: {399, 798, 1197, 1596, 1785, 1799, 1995, 2240, 2247, 2394, 2450} 50: {1065, 1070, 1424, 1425, 2130, 2140, 2475, 2490, 2495, 2498, 2550} 51: {1037, 1038, 1292, 1299, 1806, 1819, 1863, 1870, 2074, 2076, 2584, 2598, 2652} 52: {897, 900, 1422, 1430, 1794, 1800, 2379, 2380, 2678, 2691, 2700, 2702, 2756} 53: {2862} 54: {2889, 2898, 2907, 2910, 2913, 2914, 2970} 55: {670, 671, 1340, 1342, 1507, 1510, 2002, 2010, 2013, 2015, 2500, 2519, 2680, 2684, 3014, 3020, 3080} 56: {852, 854, 1248, 1253, 1704, 1708, 1869, 1876, 1878, 1880, 2496, 2506, 2556, 2562, 2835, 2848, 3108, 3122, 3128, 3129, 3132, 3134, 3192} 57: {1155, 1159, 1215, 1216, 1615, 1623, 2310, 2318, 2430, 2432, 2774, 2781, 3230, 3246, 3306} 58: {1760, 1769, 2233, 2240, 3335, 3362, 3422} 59: {3540} 60: {3570, 3580, 3585, 3588, 3590, 3594, 3595, 3596, 3597, 3598, 3660} 61: {3782} 62: {3813, 3842, 3906} 63: {1974, 1980, 1981, 1983, 2555, 2556, 2805, 2814, 3948, 3960, 3962, 3966, 4032} 64: {584, 1168, 1360, 1364, 1752, 2134, 2144, 2270, 2272, 2336, 2720, 2728, 2920, 3164, 3184, 3504, 4064, 4080, 4088, 4092, 4094, 4160} 65: {1053, 1055, 2106, 2110, 2625, 2639, 2765, 2769, 3159, 3165, 4212, 4220, 4290} 66: {869, 870, 1738, 1740, 2338, 2343, 2607, 2610, 2997, 3014, 3344, 3345, 3476, 3480, 4323, 4334, 4345, 4350, 4353, 4354, 4422} 67: {4556} 68: {4590, 4607, 4620, 4622, 4692} 69: {1674, 1679, 2369, 2379, 3348, 3358, 3564, 3565, 4738, 4758, 4830} 70: {1630, 1631, 2125, 2128, 2544, 2555, 3248, 3255, 3260, 3262, 3264, 3265, 4250, 4256, 4459, 4470, 4865, 4886, 4890, 4893, 4895, 4898, 4970} 71: {5112} 72: {5148, 5160, 5166, 5172, 5175, 5176, 5178, 5180, 5181, 5182, 5256} 73: {5402} 74: {5439, 5474, 5550} 75: {2800, 2805, 2810, 2811, 3640, 3645, 3939, 3950, 5600, 5610, 5620, 5622, 5700} 76: {1536, 1539, 1919, 1924, 2990, 3002, 3072, 3078, 3458, 3462, 3838, 3848, 4608, 4617, 4844, 4845, 5738, 5757, 5772, 5774, 5852} 77: {1869, 1870, 2959, 2961, 3738, 3740, 4422, 4431, 4444, 4445, 5607, 5610, 5918, 5922, 6006} 78: {1104, 1105, 2208, 2210, 3312, 3315, 3471, 3472, 4335, 4342, 4416, 4420, 5184, 5200, 5211, 5213, 5520, 5525, 6045, 6058, 6071, 6078, 6081, 6082, 6162} 79: {6320} 80: {6360, 6380, 6384, 6390, 6392, 6395, 6396, 6398, 6480} 81: {819, 1638, 2289, 2295, 2457, 2619, 2622, 3267, 3276, 3279, 4095, 4488, 4509, 4578, 4590, 4914, 5220, 5238, 5244, 5247, 5733, 6534, 6552, 6558, 6642} 82: {3472, 3485, 3730, 3731, 4469, 4480, 6683, 6722, 6806} 83: {6972} 84: {7014, 7028, 7035, 7042, 7044, 7049, 7050, 7052, 7053, 7054, 7140} 85: {1545, 1547, 1802, 1805, 3000, 3009, 3090, 3094, 3604, 3610, 4635, 4641, 4794, 4810, 4811, 4815, 5406, 5415, 6000, 6018, 6171, 6180, 6188, 6190, 6868, 6875, 7208, 7220, 7310} 86: {3906, 3913, 4429, 4434, 7353, 7394, 7482} 87: {3770, 3783, 5265, 5278, 7540, 7566, 7656} 88: {2574, 2580, 3200, 3201, 3990, 4004, 5148, 5152, 5159, 5160, 6400, 6402, 7700, 7722, 7733, 7736, 7740, 7742, 7832} 89: {8010} 90: {8055, 8070, 8082, 8085, 8090, 8091, 8094, 8095, 8097, 8098, 8190} 91: {1378, 1379, 2478, 2483, 2756, 2758, 3302, 3311, 3668, 3679, 4134, 4137, 4956, 4966, 5145, 5148, 5512, 5516, 6604, 6622, 6890, 6895, 7336, 7358, 7434, 7449, 8268, 8274, 8372} 92: {2415, 2416, 4550, 4554, 4830, 4832, 6480, 6509, 7245, 7248, 7797, 7808, 8418, 8441, 8460, 8462, 8556} 93: {3000, 3007, 4309, 4323, 6000, 6014, 6479, 6480, 8618, 8646, 8742} 94: {4544, 4559, 5875, 5888, 8789, 8834, 8930} 95: {4503, 4510, 5750, 5757, 9006, 9020, 9120} 96: {1840, 1842, 2420, 2424, 3680, 3684, 4840, 4848, 5508, 5520, 5526, 5528, 6279, 6304, 6783, 6784, 7260, 7272, 7360, 7368, 9168, 9184, 9192, 9200, 9204, 9208, 9210, 9212, 9213, 9214, 9312} 97: {9506} 98: {9555, 9590, 9597, 9602, 9702} 99: {2794, 2799, 3495, 3498, 4884, 4895, 4896, 4899, 5580, 5588, 5598, 5599, 5994, 5995, 6783, 6798, 6990, 6996, 8382, 8397, 9768, 9790, 9792, 9798, 9900} 100: {1110, 2220, 2724, 2725, 3325, 3330, 3332, 4235, 4240, 4440, 5134, 5150, 5448, 5450, 5550, 6640, 6650, 6660, 6664, 6665, 7756, 7770, 7775, 8145, 8172, 8175, 8180, 8470, 8480, 8880, 9950, 9975, 9980, 9990, 9995, 9996, 9998, 10100} 101: {10302} 102: {10353, 10370, 10387, 10398, 10401, 10402, 10506} 103: {10712} 104: {10764, 10790, 10803, 10808, 10812, 10814, 10920} 105: {1694, 1695, 2751, 2755, 3388, 3390, 3807, 3815, 4130, 4131, 5082, 5085, 5495, 5502, 5505, 5509, 5510, 5511, 6760, 6776, 6780, 6783, 6875, 6888, 6985, 6993, 7614, 7630, 8235, 8253, 8260, 8262, 8265, 8267, 8470, 8475, 9604, 9645, 10164, 10170, 10990, 11004, 11010, 11018, 11020, 11022, 11130} 106: {5760, 5777, 5874, 5883, 5984, 5989, 6413, 6416, 6731, 6738, 7473, 7488, 11183, 11234, 11342} 107: {11556} 108: {11610, 11628, 11637, 11646, 11652, 11655, 11658, 11660, 11661, 11662, 11772} 109: {11990} 110: {12045, 12078, 12089, 12090, 12095, 12098, 12210} 111: {4476, 4477, 4921, 4926, 6142, 6159, 8370, 8399, 8493, 8510, 8610, 8621, 8952, 8954, 9842, 9852, 12284, 12318, 12432} 112: {4172, 4176, 4179, 4180, 5080, 5082, 6422, 6440, 8344, 8352, 8358, 8360, 10160, 10164, 11165, 11184, 12488, 12516, 12528, 12530, 12536, 12537, 12540, 12542, 12656} 113: {12882} 114: {12939, 12958, 12977, 12990, 12993, 12994, 13110} 115: {2780, 2783, 5560, 5566, 6601, 6610, 8340, 8349, 8786, 8809, 8810, 8815, 11000, 11017, 11120, 11132, 13202, 13220, 13340} 116: {3504, 3509, 7008, 7018, 8062, 8070, 10512, 10527, 10752, 10759, 13398, 13427, 13452, 13454, 13572} 117: {3419, 3420, 4710, 4719, 6825, 6838, 6840, 6843, 8253, 8255, 9420, 9438, 10257, 10260, 13195, 13212, 13650, 13676, 13680, 13686, 13806} 118: {7120, 7139, 7490, 7493, 7729, 7730, 9263, 9280, 13865, 13922, 14042} 119: {7072, 7077, 8631, 8636, 14144, 14154, 14280} 120: {2055, 2056, 2538, 2540, 4110, 4112, 5076, 5080, 5915, 5928, 6156, 6165, 6168, 6170, 7614, 7620, 8220, 8224, 10152, 10160, 10250, 10275, 10280, 10284, 10972, 11010, 11830, 11856, 12312, 12320, 12330, 12336, 12339, 12340, 12690, 12700, 14340, 14360, 14370, 14376, 14380, 14385, 14388, 14390, 14392, 14394, 14395, 14396, 14397, 14398, 14520} //////////// 4. Irregular number triangle: list of numerators pertaining to qualified denominators less than or equal to b^2 + b for 2 <= b <= 120: 2: {3} 3: {4} 4: {7, 5} 5: {6} 6: {11, 17, 7} 7: {8} 8: {15, 31, 9} 9: {13, 26, 10} 10: {16, 26, 19, 49, 11} 11: {12} 12: {23, 35, 47, 71, 13} 13: {14} 14: {27, 97, 15} 15: {22, 37, 42, 57, 44, 74, 16} 16: {21, 25, 57, 42, 31, 63, 127, 17} 17: {18} 18: {35, 53, 107, 161, 19} 19: {20} 20: {39, 79, 99, 199, 21} 21: {29, 50, 31, 73, 36, 141, 58, 100, 62, 146, 22} 22: {34, 100, 58, 80, 43, 241, 23} 23: {24} 24: {47, 71, 95, 143, 191, 287, 25} 25: {31, 62, 41, 166, 93, 124, 26} 26: {67, 93, 51, 337, 27} 27: {40, 121, 74, 155, 80, 242, 28} 28: {37, 65, 43, 155, 74, 130, 55, 111, 195, 391, 29} 29: {30} 30: {59, 89, 149, 179, 299, 449, 31} 31: {32} 32: {63, 127, 255, 511, 33} 33: {45, 111, 49, 181, 90, 222, 98, 362, 34} 34: {52, 222, 90, 192, 67, 577, 35} 35: {87, 122, 129, 164, 174, 244, 36} 36: {43, 46, 82, 86, 92, 164, 129, 61, 385, 64, 172, 460, 138, 246, 71, 107, 143, 215, 323, 431, 647, 37} 37: {38} 38: {75, 721, 39} 39: {58, 253, 106, 301, 116, 506, 40} 40: {53, 133, 98, 138, 61, 301, 66, 106, 266, 426, 196, 276, 79, 159, 199, 319, 399, 799, 41} 41: {42} 42: {83, 125, 251, 293, 587, 881, 43} 43: {44} 44: {87, 175, 483, 967, 45} 45: {56, 101, 61, 196, 67, 112, 202, 337, 165, 255, 122, 392, 168, 303, 134, 224, 404, 674, 46} 46: {70, 392, 117, 255, 165, 211, 122, 352, 91, 1057, 47} 47: {48} 48: {95, 143, 191, 287, 383, 575, 767, 1151, 49} 49: {57, 114, 171, 228, 85, 771, 285, 192, 535, 342, 50} 50: {71, 321, 178, 228, 142, 642, 99, 249, 499, 1249, 51} 51: {122, 173, 76, 433, 86, 749, 138, 495, 244, 346, 152, 866, 52} 52: {69, 225, 79, 495, 138, 450, 305, 357, 103, 207, 675, 1351, 53} 53: {54} 54: {107, 161, 323, 485, 971, 1457, 55} 55: {67, 122, 134, 244, 137, 302, 91, 201, 366, 806, 100, 1145, 268, 488, 274, 604, 56} 56: {71, 183, 78, 358, 142, 366, 89, 201, 313, 705, 156, 716, 213, 549, 162, 890, 111, 223, 391, 447, 783, 1567, 57} 57: {77, 305, 135, 192, 85, 541, 154, 610, 270, 384, 219, 618, 170, 1082, 58} 58: {88, 610, 154, 560, 115, 1681, 59} 59: {60} 60: {119, 179, 239, 299, 359, 599, 719, 899, 1199, 1799, 61} 61: {62} 62: {123, 1921, 63} 63: {94, 220, 283, 661, 292, 355, 170, 737, 188, 440, 566, 1322, 64} 64: {73, 146, 85, 341, 219, 97, 737, 227, 355, 292, 170, 682, 365, 113, 1393, 438, 127, 255, 511, 1023, 2047, 65} 65: {81, 211, 162, 422, 105, 1015, 237, 497, 243, 633, 324, 844, 66} 66: {79, 145, 158, 290, 167, 497, 237, 435, 111, 1233, 380, 446, 316, 580, 131, 197, 395, 725, 1451, 2177, 67} 67: {68} 68: {135, 271, 1155, 2311, 69} 69: {93, 438, 103, 793, 186, 876, 396, 465, 206, 1586, 70} 70: {163, 233, 170, 380, 106, 876, 116, 186, 326, 466, 816, 1306, 340, 760, 273, 1043, 139, 349, 489, 699, 979, 2449, 71} 71: {72} 72: {143, 215, 287, 431, 575, 647, 863, 1295, 1727, 2591, 73} 73: {74} 74: {147, 2737, 75} 75: {112, 187, 562, 937, 273, 648, 202, 1027, 224, 374, 1124, 1874, 76} 76: {96, 324, 101, 481, 115, 1027, 192, 648, 273, 577, 202, 962, 288, 972, 519, 595, 151, 303, 1443, 2887, 77} 77: {178, 255, 269, 423, 356, 510, 134, 211, 1212, 1905, 534, 765, 538, 846, 78} 78: {92, 170, 184, 340, 276, 510, 356, 434, 289, 835, 368, 680, 144, 300, 1158, 2406, 460, 850, 155, 233, 467, 1013, 2027, 3041, 79} 79: {80} 80: {159, 319, 399, 639, 799, 1279, 1599, 3199, 81} 81: {91, 182, 109, 595, 273, 194, 437, 121, 364, 1093, 455, 136, 1837, 218, 1190, 546, 145, 388, 874, 2332, 637, 242, 728, 2186, 82} 82: {124, 1190, 373, 455, 218, 1120, 163, 3361, 83} 83: {84} 84: {167, 251, 335, 503, 587, 1007, 1175, 1763, 2351, 3527, 85} 85: {103, 273, 106, 361, 120, 885, 206, 546, 212, 722, 309, 819, 141, 481, 566, 1926, 318, 1083, 240, 1770, 242, 412, 1092, 1857, 505, 1100, 424, 1444, 86} 86: {217, 819, 309, 739, 171, 3697, 87} 87: {130, 1261, 234, 1365, 260, 2522, 88} 88: {117, 645, 300, 388, 133, 1365, 234, 322, 938, 1290, 600, 776, 175, 351, 703, 967, 1935, 3871, 89} 89: {90} 90: {179, 269, 449, 539, 809, 899, 1349, 1619, 2699, 4049, 91} 91: {106, 197, 118, 573, 212, 394, 127, 946, 131, 1132, 318, 591, 236, 1146, 420, 693, 424, 788, 254, 1892, 530, 985, 262, 2264, 354, 1719, 636, 1182, 92} 92: {210, 302, 325, 693, 420, 604, 162, 2830, 630, 906, 452, 1464, 183, 367, 2115, 4231, 93} 93: {125, 776, 139, 1441, 250, 1552, 627, 720, 278, 2882, 94} 94: {142, 1552, 250, 1472, 187, 4417, 95} 95: {237, 902, 345, 1010, 474, 1804, 96} 96: {115, 307, 121, 505, 230, 614, 242, 1010, 153, 345, 921, 2073, 161, 2561, 646, 742, 363, 1515, 460, 1228, 191, 287, 383, 575, 767, 1151, 1535, 2303, 3071, 4607, 97} 97: {98} 98: {195, 685, 1371, 4801, 99} 99: {127, 622, 233, 530, 148, 445, 544, 1633, 155, 254, 1244, 2036, 555, 654, 266, 1751, 466, 1060, 381, 1866, 296, 890, 1088, 3266, 100} 100: {111, 222, 227, 327, 133, 333, 833, 242, 742, 444, 151, 1751, 454, 654, 555, 166, 266, 666, 1666, 2666, 277, 777, 2177, 181, 681, 981, 3681, 484, 1484, 888, 199, 399, 499, 999, 1999, 2499, 4999, 101} 101: {102} 102: {203, 305, 611, 1733, 3467, 5201, 103} 103: {104} 104: {207, 415, 831, 1351, 2703, 5407, 105} 105: {121, 226, 131, 551, 242, 452, 141, 981, 354, 459, 363, 678, 157, 262, 367, 787, 1102, 1837, 169, 484, 904, 2584, 275, 1640, 381, 1221, 282, 1962, 183, 393, 708, 918, 1653, 3543, 605, 1130, 196, 4501, 726, 1356, 314, 524, 734, 1574, 2204, 3674, 106} 106: {160, 1962, 267, 1221, 374, 904, 484, 802, 381, 1123, 282, 1872, 211, 5617, 107} 107: {108} 108: {215, 323, 431, 647, 971, 1295, 1943, 2915, 3887, 5831, 109} 109: {110} 110: {219, 549, 1099, 1209, 2419, 6049, 111} 111: {373, 484, 266, 821, 166, 2053, 186, 3405, 298, 2185, 410, 1631, 746, 968, 532, 1642, 332, 4106, 112} 112: {149, 261, 597, 1045, 381, 605, 169, 2185, 298, 522, 1194, 2090, 762, 1210, 435, 2563, 223, 447, 783, 895, 1567, 1791, 3135, 6271, 113} 113: {114} 114: {227, 341, 683, 2165, 4331, 6497, 115} 115: {139, 484, 278, 968, 287, 1322, 417, 1452, 191, 766, 881, 3526, 440, 2395, 556, 1936, 574, 2644, 116} 116: {146, 726, 292, 1452, 417, 1345, 438, 2178, 672, 1484, 231, 463, 3363, 6727, 117} 117: {263, 380, 157, 1210, 175, 526, 760, 2281, 655, 889, 314, 2420, 789, 1140, 580, 2569, 350, 1052, 1520, 4562, 118} 118: {178, 2420, 535, 889, 655, 773, 314, 2320, 235, 6961, 119} 119: {416, 1011, 548, 1143, 832, 2022, 120} 120: {137, 257, 141, 381, 274, 514, 282, 762, 169, 1729, 171, 411, 771, 1851, 423, 1143, 548, 1028, 564, 1524, 205, 685, 1285, 4285, 211, 4771, 338, 3458, 342, 462, 822, 1542, 2742, 3702, 705, 1905, 239, 359, 479, 599, 719, 959, 1199, 1439, 1799, 2399, 2879, 3599, 4799, 7199, 121} //////////// 5. Algorithm: This data was produced by Wolfram language 11.1 code that required about 8 hours in both of two sessions to generate. The first session generated n/d for 2 <= b <= 120. Given the terms of the column "d" above and using that as a lower limit, the following code supplies the essential data for this chart: (The first 30 terms can be generated in minutes.) With[{bb = 120, v = Import["b292289.txt", "Data"][[All, -1]]}, Monitor[Table[ Reap[Do[If[Length@ # > 0, Sow[#], #] &@ Apply[Join, Map[{#, m} &, #]] &@ Parallelize@ Select[Range[b + 1, m - 1], Function[k, Function[{r, w, n, d}, AnyTrue[ Flatten@ Map[Apply[Outer[Divide, #1, #2] &, #] &, Transpose@ MapAt[# /. 0 -> Nothing &, Map[Function[x, Map[Map[FromDigits[#, b] &@ Delete[x, #] &, Position[x, #]] &, Intersection @@ {n, d}]], {n, d}], -1]], # == Divide @@ {k, m} &]] @@ {k/m, #, First@ #, Last@ #} &@Map[IntegerDigits[#, b] &, {k, m}] - Boole[Mod[{k, m}, b] == {0, 0}]] ], {m, v[[b - 1]], b^2 + b}]][[-1, -1]], {b, 2, bb}], ProgressIndicator[b, {2, bb}]]] (eof)