User:Aadenboy/Self-equaling squares/d^3

Self-equaling cubes in base-N. 0 and 1 are still constants, but interestingly, N-1 is also a constant.

Bases 1 to 100

Base	Total	Digits
2	2	0, 1
3	3	0, 1, 2
4	3	0, 1, 3
5	3	0, 1, 4
6	6	0, 1, 2, 3, 4, 5
7	3	0, 1, 6
8	5	0, 1, 3, 5, 7
9	3	0, 1, 8
10	6	0, 1, 4, 5, 6, 9
11	3	0, 1, 10
12	9	0, 1, 3, 4, 5, 7, 8, 9, 11
13	3	0, 1, 12
14	6	0, 1, 6, 7, 8, 13
15	9	0, 1, 4, 5, 6, 9, 10, 11, 14
16	5	0, 1, 7, 9, 15
17	3	0, 1, 16
18	6	0, 1, 8, 9, 10, 17
19	3	0, 1, 18
20	9	0, 1, 4, 5, 9, 11, 15, 16, 19
21	9	0, 1, 6, 7, 8, 13, 14, 15, 20
22	6	0, 1, 10, 11, 12, 21
23	3	0, 1, 22
24	15	0, 1, 3, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23
25	3	0, 1, 24
26	6	0, 1, 12, 13, 14, 25
27	3	0, 1, 26
28	9	0, 1, 7, 8, 13, 15, 20, 21, 27
29	3	0, 1, 28
30	18	0, 1, 4, 5, 6, 9, 10, 11, 14, 15, 16, 19, 20, 21, 24, 25, 26, 29
31	3	0, 1, 30
32	5	0, 1, 15, 17, 31
33	9	0, 1, 10, 11, 12, 21, 22, 23, 32
34	6	0, 1, 16, 17, 18, 33
35	9	0, 1, 6, 14, 15, 20, 21, 29, 34
36	9	0, 1, 8, 9, 17, 19, 27, 28, 35
37	3	0, 1, 36
38	6	0, 1, 18, 19, 20, 37
39	9	0, 1, 12, 13, 14, 25, 26, 27, 38
40	15	0, 1, 5, 9, 11, 15, 16, 19, 21, 24, 25, 29, 31, 35, 39
41	3	0, 1, 40
42	18	0, 1, 6, 7, 8, 13, 14, 15, 20, 21, 22, 27, 28, 29, 34, 35, 36, 41
43	3	0, 1, 42
44	9	0, 1, 11, 12, 21, 23, 32, 33, 43
45	9	0, 1, 9, 10, 19, 26, 35, 36, 44
46	6	0, 1, 22, 23, 24, 45
47	3	0, 1, 46
48	15	0, 1, 7, 9, 15, 16, 17, 23, 25, 31, 32, 33, 39, 41, 47
49	3	0, 1, 48
50	6	0, 1, 24, 25, 26, 49
51	9	0, 1, 16, 17, 18, 33, 34, 35, 50
52	9	0, 1, 12, 13, 25, 27, 39, 40, 51
53	3	0, 1, 52
54	6	0, 1, 26, 27, 28, 53
55	9	0, 1, 10, 11, 21, 34, 44, 45, 54
56	15	0, 1, 7, 8, 13, 15, 21, 27, 29, 35, 41, 43, 48, 49, 55
57	9	0, 1, 18, 19, 20, 37, 38, 39, 56
58	6	0, 1, 28, 29, 30, 57
59	3	0, 1, 58
60	27	0, 1, 4, 5, 9, 11, 15, 16, 19, 20, 21, 24, 25, 29, 31, 35, 36, 39, 40, 41, 44, 45, 49, 51, 55, 56, 59
61	3	0, 1, 60
62	6	0, 1, 30, 31, 32, 61
63	9	0, 1, 8, 27, 28, 35, 36, 55, 62
64	5	0, 1, 31, 33, 63
65	9	0, 1, 14, 25, 26, 39, 40, 51, 64
66	18	0, 1, 10, 11, 12, 21, 22, 23, 32, 33, 34, 43, 44, 45, 54, 55, 56, 65
67	3	0, 1, 66
68	9	0, 1, 16, 17, 33, 35, 51, 52, 67
69	9	0, 1, 22, 23, 24, 45, 46, 47, 68
70	18	0, 1, 6, 14, 15, 20, 21, 29, 34, 35, 36, 41, 49, 50, 55, 56, 64, 69
71	3	0, 1, 70
72	15	0, 1, 8, 9, 17, 19, 27, 35, 37, 45, 53, 55, 63, 64, 71
73	3	0, 1, 72
74	6	0, 1, 36, 37, 38, 73
75	9	0, 1, 24, 25, 26, 49, 50, 51, 74
76	9	0, 1, 19, 20, 37, 39, 56, 57, 75
77	9	0, 1, 21, 22, 34, 43, 55, 56, 76
78	18	0, 1, 12, 13, 14, 25, 26, 27, 38, 39, 40, 51, 52, 53, 64, 65, 66, 77
79	3	0, 1, 78
80	15	0, 1, 9, 15, 16, 25, 31, 39, 41, 49, 55, 64, 65, 71, 79
81	3	0, 1, 80
82	6	0, 1, 40, 41, 42, 81
83	3	0, 1, 82
84	27	0, 1, 7, 8, 13, 15, 20, 21, 27, 28, 29, 35, 36, 41, 43, 48, 49, 55, 56, 57, 63, 64, 69, 71, 76, 77, 83
85	9	0, 1, 16, 34, 35, 50, 51, 69, 84
86	6	0, 1, 42, 43, 44, 85
87	9	0, 1, 28, 29, 30, 57, 58, 59, 86
88	15	0, 1, 11, 21, 23, 32, 33, 43, 45, 55, 56, 65, 67, 77, 87
89	3	0, 1, 88
90	18	0, 1, 9, 10, 19, 26, 35, 36, 44, 45, 46, 54, 55, 64, 71, 80, 81, 89
91	9	0, 1, 13, 14, 27, 64, 77, 78, 90
92	9	0, 1, 23, 24, 45, 47, 68, 69, 91
93	9	0, 1, 30, 31, 32, 61, 62, 63, 92
94	6	0, 1, 46, 47, 48, 93
95	9	0, 1, 19, 20, 39, 56, 75, 76, 94
96	15	0, 1, 15, 17, 31, 32, 33, 47, 49, 63, 64, 65, 79, 81, 95
97	3	0, 1, 96
98	6	0, 1, 48, 49, 50, 97
99	9	0, 1, 10, 44, 45, 54, 55, 89, 98
100	9	0, 1, 24, 25, 49, 51, 75, 76, 99

Record breakers

Base	Total	Digits
2	2	0, 1
3	3	0, 1, 2
6	6	0, 1, 2, 3, 4, 5
12	9	0, 1, 3, 4, 5, 7, 8, 9, 11
24	15	0, 1, 3, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23
30	18	0, 1, 4, 5, 6, 9, 10, 11, 14, 15, 16, 19, 20, 21, 24, 25, 26, 29
60	27	0, 1, 4, 5, 9, 11, 15, 16, 19, 20, 21, 24, 25, 29, 31, 35, 36, 39, 40, 41, 44, 45, 49, 51, 55, 56, 59
120	45	0, 1, 5, 9, 11, 15, 16, 19, 21, 24, 25, 29, 31, 35, 39, 40, 41, 45, 49, 51, 55, 56, 59, 61, 64, 65, 69, 71, 75, 79, 80, 81, 85, 89, 91, 95, 96, 99, 101, 104, 105, 109, 111, 115, 119
210	54	0, 1, 6, 14, 15, 20, 21, 29, 34, 35, 36, 41, 49, 50, 55, 56, 64, 69, 70, 71, 76, 84, 85, 90, 91, 99, 104, 105, 106, 111, 119, 120, 125, 126, 134, 139, 140, 141, 146, 154, 155, 160, 161, 169, 174, 175, 176, 181, 189, 190, 195, 196, 204, 209
420	81	0, 1, 15, 20, 21, 29, 35, 36, 41, 49, 55, 56, 64, 69, 71, 76, 84, 85, 91, 99, 104, 105, 111, 119, 120, 125, 139, 140, 141, 155, 160, 161, 169, 175, 176, 181, 189, 195, 196, 204, 209, 211, 216, 224, 225, 231, 239, 244, 245, 251, 259, 260, 265, 279, 280, 281, 295, 300, 301, 309, 315, 316, 321, 329, 335, 336, 344, 349, 351, 356, 364, 365, 371, 379, 384, 385, 391, 399, 400, 405, 419
840	135	0, 1, 15, 21, 29, 35, 41, 49, 55, 56, 64, 69, 71, 85, 91, 99, 104, 105, 111, 119, 120, 125, 139, 141, 155, 160, 161, 169, 175, 176, 181, 189, 195, 209, 211, 216, 224, 225, 231, 239, 245, 251, 259, 265, 279, 280, 281, 295, 301, 309, 315, 321, 329, 335, 336, 344, 349, 351, 365, 371, 379, 384, 385, 391, 399, 400, 405, 419, 421, 435, 440, 441, 449, 455, 456, 461, 469, 475, 489, 491, 496, 504, 505, 511, 519, 525, 531, 539, 545, 559, 560, 561, 575, 581, 589, 595, 601, 609, 615, 616, 624, 629, 631, 645, 651, 659, 664, 665, 671, 679, 680, 685, 699, 701, 715, 720, 721, 729, 735, 736, 741, 749, 755, 769, 771, 776, 784, 785, 791, 799, 805, 811, 819, 825, 839
2310	162	0, 1, 21, 34, 55, 56, 76, 99, 111, 120, 154, 155, 175, 176, 209, 210, 230, 231, 265, 274, 286, 309, 329, 330, 351, 364, 384, 385, 386, 406, 419, 440, 441, 461, 484, 496, 505, 539, 540, 560, 561, 594, 595, 615, 616, 650, 659, 671, 694, 714, 715, 736, 749, 769, 770, 771, 791, 804, 825, 826, 846, 869, 881, 890, 924, 925, 945, 946, 979, 980, 1000, 1001, 1035, 1044, 1056, 1079, 1099, 1100, 1121, 1134, 1154, 1155, 1156, 1176, 1189, 1210, 1211, 1231, 1254, 1266, 1275, 1309, 1310, 1330, 1331, 1364, 1365, 1385, 1386, 1420, 1429, 1441, 1464, 1484, 1485, 1506, 1519, 1539, 1540, 1541, 1561, 1574, 1595, 1596, 1616, 1639, 1651, 1660, 1694, 1695, 1715, 1716, 1749, 1750, 1770, 1771, 1805, 1814, 1826, 1849, 1869, 1870, 1891, 1904, 1924, 1925, 1926, 1946, 1959, 1980, 1981, 2001, 2024, 2036, 2045, 2079, 2080, 2100, 2101, 2134, 2135, 2155, 2156, 2190, 2199, 2211, 2234, 2254, 2255, 2276, 2289, 2309
4620	243	0, 1, 21, 55, 56, 76, 99, 111, 120, 155, 175, 176, 209, 231, 265, 309, 329, 351, 364, 384, 385, 419, 440, 441, 461, 484, 496, 505, 539, 540, 560, 561, 595, 615, 616, 659, 671, 715, 736, 749, 769, 771, 791, 804, 825, 869, 881, 924, 925, 945, 979, 980, 1000, 1001, 1035, 1044, 1056, 1079, 1099, 1100, 1121, 1155, 1156, 1176, 1189, 1211, 1231, 1275, 1309, 1331, 1364, 1365, 1385, 1420, 1429, 1441, 1464, 1484, 1485, 1519, 1539, 1540, 1541, 1561, 1595, 1596, 1616, 1639, 1651, 1660, 1695, 1715, 1716, 1749, 1771, 1805, 1849, 1869, 1891, 1904, 1924, 1925, 1959, 1980, 1981, 2001, 2024, 2036, 2045, 2079, 2080, 2100, 2101, 2135, 2155, 2156, 2199, 2211, 2255, 2276, 2289, 2309, 2311, 2331, 2344, 2365, 2409, 2421, 2464, 2465, 2485, 2519, 2520, 2540, 2541, 2575, 2584, 2596, 2619, 2639, 2640, 2661, 2695, 2696, 2716, 2729, 2751, 2771, 2815, 2849, 2871, 2904, 2905, 2925, 2960, 2969, 2981, 3004, 3024, 3025, 3059, 3079, 3080, 3081, 3101, 3135, 3136, 3156, 3179, 3191, 3200, 3235, 3255, 3256, 3289, 3311, 3345, 3389, 3409, 3431, 3444, 3464, 3465, 3499, 3520, 3521, 3541, 3564, 3576, 3585, 3619, 3620, 3640, 3641, 3675, 3695, 3696, 3739, 3751, 3795, 3816, 3829, 3849, 3851, 3871, 3884, 3905, 3949, 3961, 4004, 4005, 4025, 4059, 4060, 4080, 4081, 4115, 4124, 4136, 4159, 4179, 4180, 4201, 4235, 4236, 4256, 4269, 4291, 4311, 4355, 4389, 4411, 4444, 4445, 4465, 4500, 4509, 4521, 4544, 4564, 4565, 4599, 4619
9240	405	0, 1, 21, 55, 56, 99, 111, 120, 155, 175, 176, 209, 231, 265, 309, 329, 351, 384, 385, 419, 440, 441, 461, 496, 505, 539, 560, 561, 595, 615, 616, 659, 671, 715, 736, 749, 769, 771, 791, 825, 869, 881, 925, 945, 979, 1000, 1001, 1035, 1056, 1079, 1099, 1121, 1155, 1176, 1189, 1211, 1231, 1275, 1309, 1331, 1365, 1385, 1429, 1441, 1464, 1485, 1519, 1539, 1541, 1561, 1595, 1616, 1639, 1651, 1695, 1715, 1749, 1771, 1805, 1849, 1869, 1891, 1904, 1925, 1959, 1981, 2001, 2024, 2045, 2079, 2080, 2101, 2135, 2155, 2199, 2211, 2255, 2289, 2309, 2311, 2331, 2344, 2365, 2409, 2421, 2464, 2465, 2485, 2519, 2520, 2541, 2575, 2584, 2619, 2639, 2640, 2661, 2695, 2696, 2729, 2751, 2771, 2815, 2849, 2871, 2904, 2905, 2925, 2960, 2969, 2981, 3024, 3025, 3059, 3079, 3080, 3081, 3101, 3135, 3136, 3179, 3191, 3200, 3235, 3255, 3256, 3289, 3311, 3345, 3389, 3409, 3431, 3464, 3465, 3499, 3520, 3521, 3541, 3576, 3585, 3619, 3640, 3641, 3675, 3695, 3696, 3739, 3751, 3795, 3816, 3829, 3849, 3851, 3871, 3905, 3949, 3961, 4005, 4025, 4059, 4080, 4081, 4115, 4136, 4159, 4179, 4201, 4235, 4256, 4269, 4291, 4311, 4355, 4389, 4411, 4445, 4465, 4509, 4521, 4544, 4565, 4599, 4619, 4621, 4641, 4675, 4696, 4719, 4731, 4775, 4795, 4829, 4851, 4885, 4929, 4949, 4971, 4984, 5005, 5039, 5061, 5081, 5104, 5125, 5159, 5160, 5181, 5215, 5235, 5279, 5291, 5335, 5369, 5389, 5391, 5411, 5424, 5445, 5489, 5501, 5544, 5545, 5565, 5599, 5600, 5621, 5655, 5664, 5699, 5719, 5720, 5741, 5775, 5776, 5809, 5831, 5851, 5895, 5929, 5951, 5984, 5985, 6005, 6040, 6049, 6061, 6104, 6105, 6139, 6159, 6160, 6161, 6181, 6215, 6216, 6259, 6271, 6280, 6315, 6335, 6336, 6369, 6391, 6425, 6469, 6489, 6511, 6544, 6545, 6579, 6600, 6601, 6621, 6656, 6665, 6699, 6720, 6721, 6755, 6775, 6776, 6819, 6831, 6875, 6896, 6909, 6929, 6931, 6951, 6985, 7029, 7041, 7085, 7105, 7139, 7160, 7161, 7195, 7216, 7239, 7259, 7281, 7315, 7336, 7349, 7371, 7391, 7435, 7469, 7491, 7525, 7545, 7589, 7601, 7624, 7645, 7679, 7699, 7701, 7721, 7755, 7776, 7799, 7811, 7855, 7875, 7909, 7931, 7965, 8009, 8029, 8051, 8064, 8085, 8119, 8141, 8161, 8184, 8205, 8239, 8240, 8261, 8295, 8315, 8359, 8371, 8415, 8449, 8469, 8471, 8491, 8504, 8525, 8569, 8581, 8624, 8625, 8645, 8679, 8680, 8701, 8735, 8744, 8779, 8799, 8800, 8821, 8855, 8856, 8889, 8911, 8931, 8975, 9009, 9031, 9064, 9065, 9085, 9120, 9129, 9141, 9184, 9185, 9219, 9239
30030	486	0, 1, 155, 209, 274, 286, 351, 364, 441, 560, 650, 714, 715, 846, 924, 1000, 1001, 1079, 1156, 1210, 1275, 1364, 1365, 1429, 1561, 1574, 1639, 1651, 1715, 1716, 1924, 1925, 2001, 2079, 2080, 2211, 2276, 2289, 2365, 2366, 2430, 2575, 2639, 2640, 2716, 2729, 2794, 2925, 2926, 3004, 3080, 3081, 3289, 3290, 3354, 3366, 3431, 3444, 3576, 3640, 3641, 3730, 3795, 3849, 3926, 4004, 4005, 4081, 4159, 4290, 4291, 4355, 4445, 4564, 4641, 4654, 4719, 4731, 4796, 4850, 5004, 5005, 5006, 5160, 5214, 5279, 5291, 5356, 5369, 5446, 5565, 5655, 5719, 5720, 5851, 5929, 6005, 6006, 6084, 6161, 6215, 6280, 6369, 6370, 6434, 6566, 6579, 6644, 6656, 6720, 6721, 6929, 6930, 7006, 7084, 7085, 7216, 7281, 7294, 7370, 7371, 7435, 7580, 7644, 7645, 7721, 7734, 7799, 7930, 7931, 8009, 8085, 8086, 8294, 8295, 8359, 8371, 8436, 8449, 8581, 8645, 8646, 8735, 8800, 8854, 8931, 9009, 9010, 9086, 9164, 9295, 9296, 9360, 9450, 9569, 9646, 9659, 9724, 9736, 9801, 9855, 10009, 10010, 10011, 10165, 10219, 10284, 10296, 10361, 10374, 10451, 10570, 10660, 10724, 10725, 10856, 10934, 11010, 11011, 11089, 11166, 11220, 11285, 11374, 11375, 11439, 11571, 11584, 11649, 11661, 11725, 11726, 11934, 11935, 12011, 12089, 12090, 12221, 12286, 12299, 12375, 12376, 12440, 12585, 12649, 12650, 12726, 12739, 12804, 12935, 12936, 13014, 13090, 13091, 13299, 13300, 13364, 13376, 13441, 13454, 13586, 13650, 13651, 13740, 13805, 13859, 13936, 14014, 14015, 14091, 14169, 14300, 14301, 14365, 14455, 14574, 14651, 14664, 14729, 14741, 14806, 14860, 15014, 15015, 15016, 15170, 15224, 15289, 15301, 15366, 15379, 15456, 15575, 15665, 15729, 15730, 15861, 15939, 16015, 16016, 16094, 16171, 16225, 16290, 16379, 16380, 16444, 16576, 16589, 16654, 16666, 16730, 16731, 16939, 16940, 17016, 17094, 17095, 17226, 17291, 17304, 17380, 17381, 17445, 17590, 17654, 17655, 17731, 17744, 17809, 17940, 17941, 18019, 18095, 18096, 18304, 18305, 18369, 18381, 18446, 18459, 18591, 18655, 18656, 18745, 18810, 18864, 18941, 19019, 19020, 19096, 19174, 19305, 19306, 19370, 19460, 19579, 19656, 19669, 19734, 19746, 19811, 19865, 20019, 20020, 20021, 20175, 20229, 20294, 20306, 20371, 20384, 20461, 20580, 20670, 20734, 20735, 20866, 20944, 21020, 21021, 21099, 21176, 21230, 21295, 21384, 21385, 21449, 21581, 21594, 21659, 21671, 21735, 21736, 21944, 21945, 22021, 22099, 22100, 22231, 22296, 22309, 22385, 22386, 22450, 22595, 22659, 22660, 22736, 22749, 22814, 22945, 22946, 23024, 23100, 23101, 23309, 23310, 23374, 23386, 23451, 23464, 23596, 23660, 23661, 23750, 23815, 23869, 23946, 24024, 24025, 24101, 24179, 24310, 24311, 24375, 24465, 24584, 24661, 24674, 24739, 24751, 24816, 24870, 25024, 25025, 25026, 25180, 25234, 25299, 25311, 25376, 25389, 25466, 25585, 25675, 25739, 25740, 25871, 25949, 26025, 26026, 26104, 26181, 26235, 26300, 26389, 26390, 26454, 26586, 26599, 26664, 26676, 26740, 26741, 26949, 26950, 27026, 27104, 27105, 27236, 27301, 27314, 27390, 27391, 27455, 27600, 27664, 27665, 27741, 27754, 27819, 27950, 27951, 28029, 28105, 28106, 28314, 28315, 28379, 28391, 28456, 28469, 28601, 28665, 28666, 28755, 28820, 28874, 28951, 29029, 29030, 29106, 29184, 29315, 29316, 29380, 29470, 29589, 29666, 29679, 29744, 29756, 29821, 29875, 30029