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Posted (edited)

A question.

 

Is there any mathematical significance to the fact that the prime number 17 has a cube 17^3 = 4913, and then the sum of these numbers (4+9+1+3) = 17 ?

 

Then for NON PRIME number 27, the cube 27^3 = 19683, and again the sum of the numbers (1+9+6+8+3) = 27. The pattern does not fit for 37^3 = 50653

 

Is this cubing pattern found in any other numbers ?

 

==

 

Edit:

Found another, number 18. 18^3 = 5832; (5+8+3+2) = 18

 

Also number 8. 8^3 = 512. (5+1+2) = 8

 

So, at this time the pattern holds for prime number 17 and non prime numbers 8, 18, 27.

Edited by Rade
Posted

well, to put it bluntly there is no significance to that pattern.

for example, if i wrote 17^3 in base 16, that would be

4913 | 16

307 | 1

19 | 3

1 | 3

0 | 1

 

1331, which when i add the hexes, gives 8.

 

there are probably numbers in base 16 that when cubed and added give the same value.

Posted

well, to put it bluntly there is no significance to that pattern.

 

 

I bluntly agree to the degree that 'significance' as used by Rade may mean some deep or profound utility or application. Rade; will you clarify this for us?

for example, if i wrote 17^3 in base 16, that would be

4913 | 16

307 | 1

19 | 3

1 | 3

0 | 1

 

1331, which when i add the hexes, gives 8.

 

there are probably numbers in base 16 that when cubed and added give the same value.

 

Exactly! This goes to the point of not making the mistake of confusing number with numertion. [Notwithstanding that I haven't checked your work.]

 

Nonetheless I think Rade poses an interesting recreational mathematical sequence in base 10 and that Phillip the natural extension of it to all bases. Phillip or Craig et al can handily and happily fill in for say base 2 to base 16 or such a matter. Perform exhaustive program searches for each base up to say 1000000 to start and record/count the results. How many cubes have digital sums that equals the base of the exponent 3 and what are they? With complete lists of sufficient size, we might look for generalizing expressions. :clue:

 

It may be the case that these sequences have been discovered, but I'm unsure how to search such a thing. :reallyconfused:

Posted (edited)
I bluntly agree to the degree that 'significance' as used by Rade may mean some deep or profound utility or application. Rade; will you clarify this for us?

I read of the pattern for the prime number 17, then made some quick calculations and identified the same for non prime numbers 8, 18, 27. I do not have any utility or application in mind.

 

Edit: I see from this link that the numbers 8 and 27 from the (8,17,18,27) pattern noted above are 'cubic numbers' related to figurate numbers:

 

http://mathworld.wolfram.com/CubicNumber.html

Edited by Rade
Posted

I agree with Phillip and Turtle about the lack of “significance” or “deep or profound utility or application”. I’d add my own maxim about “numerological” functions, such as summing the digits of fixed base system number: if the remarkable coincidence you’re seeing doesn’t persist in many bases, it’s a coincidence of base system you’ve selected (or defaulted to, which is usually the case with base 10).

 

Here’s a heap of all the numbers A where the sum of the digits of their base B representation of their cube sums to them, for bases 2 through 100:

B count A A^3
2 1 1 1
3 1 1 1
3 2 4 64
3 3 8 512
4 1 1 1
4 2 2 8
4 3 6 216
4 4 7 343
4 5 9 729
4 6 10 1000
5 1 1 1
5 2 3 27
5 3 8 512
5 4 9 729
5 5 12 1728
5 6 13 2197
6 1 1 1
6 2 9 729
6 3 11 1331
6 4 15 3375
6 5 16 4096
7 1 1 1
7 2 2 8
7 3 4 64
7 4 8 512
7 5 9 729
7 6 11 1331
7 7 12 1728
7 8 15 3375
7 9 16 4096
8 1 1 1
8 2 6 216
8 3 13 2197
8 4 14 2744
9 1 1 1
9 2 3 27
9 3 7 343
9 4 15 3375
9 5 16 4096
9 6 23 12167
9 7 32 32768
10 1 1 1
10 2 8 512
10 3 17 4913
10 4 18 5832
10 5 26 17576
10 6 27 19683
11 1 1 1
11 2 5 125
11 3 9 729
11 4 14 2744
11 5 16 4096
11 6 19 6859
11 7 24 13824
12 1 1 1
12 2 21 9261
12 3 22 10648
12 4 23 12167
12 5 32 32768
12 6 33 35937
12 7 34 39304
13 1 1 1
13 2 3 27
13 3 7 343
13 4 8 512
13 5 9 729
13 6 16 4096
13 7 17 4913
13 8 19 6859
13 9 20 8000
13 10 21 9261
13 11 24 13824
13 12 25 15625
13 13 27 19683
13 14 32 32768
14 1 1 1
14 2 25 15625
14 3 26 17576
14 4 27 19683
14 5 38 54872
15 1 1 1
15 2 8 512
15 3 20 8000
15 4 21 9261
15 5 22 10648
15 6 27 19683
15 7 28 21952
15 8 29 24389
15 9 34 39304
15 10 35 42875
15 11 36 46656
16 1 1 1
16 2 4 64
16 3 11 1331
16 4 20 8000
16 5 21 9261
16 6 25 15625
16 7 26 17576
16 8 29 24389
16 9 30 27000
16 10 31 29791
16 11 34 39304
16 12 35 42875
16 13 39 59319
17 1 1 1
17 2 7 343
17 3 23 12167
17 4 32 32768
17 5 33 35937
17 6 55 166375
18 1 1 1
18 2 33 35937
18 3 34 39304
18 4 35 42875
19 1 1 1
19 2 9 729
19 3 26 17576
19 4 28 21952
19 5 35 42875
19 6 36 46656
19 7 37 50653
19 8 44 85184
19 9 46 97336
20 1 1 1
20 2 37 50653
20 3 38 54872
20 4 39 59319
21 1 1 1
21 2 4 64
21 3 11 1331
21 4 16 4096
21 5 24 13824
21 6 25 15625
21 7 29 24389
21 8 31 29791
21 9 35 42875
21 10 36 46656
21 11 39 59319
21 12 40 64000
21 13 41 68921
21 14 45 91125
21 15 49 117649
21 16 51 132651
21 17 55 166375
21 18 56 175616
22 1 1 1
22 2 8 512
22 3 28 21952
22 4 29 24389
22 5 34 39304
22 6 35 42875
22 7 36 46656
22 8 41 68921
22 9 42 74088
22 10 43 79507
22 11 48 110592
22 12 50 125000
22 13 55 166375
22 14 56 175616
22 15 57 185193
23 1 1 1
23 2 12 1728
23 3 34 39304
23 4 43 79507
23 5 45 91125
23 6 54 157464
23 7 55 166375
23 8 56 175616
24 1 1 1
24 2 45 91125
24 3 46 97336
24 4 47 103823
24 5 68 314432
24 6 69 328509
25 1 1 1
25 2 3 27
25 3 5 125
25 4 9 729
25 5 11 1331
25 6 15 3375
25 7 27 19683
25 8 29 24389
25 9 32 32768
25 10 33 35937
25 11 35 42875
25 12 41 68921
25 13 45 91125
25 14 47 103823
25 15 48 110592
25 16 49 117649
25 17 56 175616
25 18 57 185193
25 19 64 262144
25 20 71 357911
25 21 72 373248
25 22 73 389017
26 1 1 1
26 2 49 117649
26 3 50 125000
26 4 51 132651
26 5 76 438976
27 1 1 1
27 2 12 1728
27 3 13 2197
27 4 38 54872
27 5 40 64000
27 6 51 132651
27 7 52 140608
27 8 53 148877
27 9 65 274625
28 1 1 1
28 2 53 148877
28 3 54 157464
28 4 55 166375
29 1 1 1
29 2 15 3375
29 3 21 9261
29 4 43 79507
29 5 48 110592
29 6 55 166375
29 7 56 175616
29 8 57 185193
29 9 64 262144
29 10 69 328509
29 11 71 357911
29 12 76 438976
29 13 77 456533
29 14 83 571787
30 1 1 1
30 2 57 185193
30 3 58 195112
30 4 59 205379
30 5 86 636056
31 1 1 1
31 2 4 64
31 3 5 125
31 4 10 1000
31 5 16 4096
31 6 19 6859
31 7 20 8000
31 8 25 15625
31 9 35 42875
31 10 36 46656
31 11 40 64000
31 12 41 68921
31 13 45 91125
31 14 46 97336
31 15 49 117649
31 16 54 157464
31 17 55 166375
31 18 56 175616
31 19 59 205379
31 20 60 216000
31 21 61 226981
31 22 64 262144
31 23 65 274625
31 24 70 343000
31 25 74 405224
31 26 75 421875
31 27 76 438976
31 28 84 592704
31 29 86 636056
31 30 89 704969
31 31 96 884736
32 1 1 1
32 2 61 226981
32 3 62 238328
32 4 63 250047
33 1 1 1
33 2 15 3375
33 3 47 103823
33 4 63 250047
33 5 64 262144
33 6 65 274625
33 7 79 493039
34 1 1 1
34 2 11 1331
34 3 21 9261
34 4 22 10648
34 5 43 79507
34 6 44 85184
34 7 45 91125
34 8 55 166375
34 9 56 175616
34 10 65 274625
34 11 66 287496
34 12 67 300763
34 13 77 456533
34 14 87 658503
34 15 89 704969
35 1 1 1
35 2 16 4096
35 3 17 4913
35 4 50 125000
35 5 52 140608
35 6 68 314432
35 7 69 328509
35 8 84 592704
36 1 1 1
36 2 6 216
36 3 14 2744
36 4 20 8000
36 5 21 9261
36 6 41 68921
36 7 49 117649
36 8 50 125000
36 9 55 166375
36 10 56 175616
36 11 64 262144
36 12 70 343000
36 13 71 357911
36 14 76 438976
36 15 99 970299
36 16 106 1191016
37 1 1 1
37 2 19 6859
37 3 28 21952
37 4 44 85184
37 5 55 166375
37 6 63 250047
37 7 64 262144
37 8 72 373248
37 9 73 389017
37 10 80 512000
37 11 89 704969
37 12 91 753571
37 13 109 1295029
37 14 117 1601613
38 1 1 1
38 2 74 405224
38 3 75 421875
38 4 112 1404928
39 1 1 1
39 2 20 8000
39 3 57 185193
39 4 58 195112
39 5 76 438976
39 6 77 456533
39 7 95 857375
39 8 96 884736
39 9 115 1520875
40 1 1 1
40 2 12 1728
40 3 27 19683
40 4 51 132651
40 5 52 140608
40 6 53 148877
40 7 64 262144
40 8 65 274625
40 9 66 287496
40 10 78 474552
40 11 79 493039
40 12 91 753571
40 13 118 1643032
41 1 1 1
41 2 5 125
41 3 15 3375
41 4 19 6859
41 5 24 13824
41 6 25 15625
41 7 45 91125
41 8 56 175616
41 9 59 205379
41 10 65 274625
41 11 69 328509
41 12 71 357911
41 13 75 421875
41 14 80 512000
41 15 81 531441
41 16 85 614125
41 17 91 753571
41 18 96 884736
41 19 101 1030301
41 20 105 1157625
41 21 111 1367631
41 22 121 1771561
41 23 139 2685619
42 1 1 1
42 2 82 551368
42 3 83 571787
42 4 124 1906624
43 1 1 1
43 2 6 216
43 3 13 2197
43 4 20 8000
43 5 21 9261
43 6 29 24389
43 7 34 39304
43 8 35 42875
43 9 36 46656
43 10 49 117649
43 11 55 166375
43 12 56 175616
43 13 57 185193
43 14 64 262144
43 15 69 328509
43 16 70 343000
43 17 71 357911
43 18 76 438976
43 19 77 456533
43 20 78 474552
43 21 84 592704
43 22 85 614125
43 23 91 753571
43 24 97 912673
43 25 112 1404928
43 26 127 2048383
43 27 132 2299968
43 28 147 3176523
44 1 1 1
44 2 86 636056
44 3 87 658503
44 4 130 2197000
45 1 1 1
45 2 23 12167
45 3 32 32768
45 4 55 166375
45 5 65 274625
45 6 67 300763
45 7 76 438976
45 8 88 681472
45 9 89 704969
45 10 100 1000000
45 11 133 2352637
45 12 155 3723875
46 1 1 1
46 2 19 6859
46 3 26 17576
46 4 35 42875
46 5 36 46656
46 6 54 157464
46 7 71 357911
46 8 80 512000
46 9 90 729000
46 10 91 753571
46 11 99 970299
46 12 136 2515456
47 1 1 1
47 2 24 13824
47 3 69 328509
47 4 70 343000
47 5 92 778688
47 6 93 804357
47 7 139 2685619
48 1 1 1
48 2 94 830584
48 3 95 857375
48 4 140 2744000
48 5 142 2863288
49 1 1 1
49 2 7 343
49 3 17 4913
49 4 23 12167
49 5 55 166375
49 6 57 185193
49 7 64 262144
49 8 65 274625
49 9 80 512000
49 10 81 531441
49 11 87 658503
49 12 89 704969
49 13 96 884736
49 14 97 912673
49 15 111 1367631
49 16 113 1442897
49 17 135 2460375
49 18 137 2571353
49 19 143 2924207
49 20 145 3048625
49 21 160 4096000
50 1 1 1
50 2 98 941192
50 3 99 970299
50 4 146 3112136
50 5 148 3241792
51 1 1 1
51 2 24 13824
51 3 25 15625
51 4 74 405224
51 5 76 438976
51 6 100 1000000
51 7 101 1030301
51 8 125 1953125
51 9 149 3307949
51 10 151 3442951
52 1 1 1
52 2 18 5832
52 3 33 35937
52 4 67 300763
52 5 68 314432
52 6 69 328509
52 7 86 636056
52 8 102 1061208
52 9 103 1092727
52 10 118 1643032
52 11 119 1685159
52 12 135 2460375
52 13 152 3511808
52 14 154 3652264
52 15 170 4913000
53 1 1 1
53 2 27 19683
53 3 39 59319
53 4 64 262144
53 5 77 456533
53 6 79 493039
53 7 104 1124864
53 8 105 1157625
53 9 116 1560896
53 10 157 3869893
53 11 181 5929741
54 1 1 1
54 2 105 1157625
54 3 106 1191016
54 4 107 1225043
55 1 1 1
55 2 28 21952
55 3 81 531441
55 4 82 551368
55 5 107 1225043
55 6 108 1259712
55 7 109 1295029
56 1 1 1
56 2 44 85184
56 3 45 91125
56 4 65 274625
56 5 89 704969
56 6 99 970299
56 7 100 1000000
56 8 109 1295029
56 9 110 1331000
56 10 111 1367631
56 11 131 2248091
56 12 199 7880599
57 1 1 1
57 2 7 343
57 3 15 3375
57 4 27 19683
57 5 35 42875
57 6 41 68921
57 7 48 110592
57 8 49 117649
57 9 64 262144
57 10 69 328509
57 11 71 357911
57 12 77 456533
57 13 83 571787
57 14 91 753571
57 15 99 970299
57 16 111 1367631
57 17 112 1404928
57 18 113 1442897
57 19 120 1728000
58 1 1 1
58 2 19 6859
58 3 37 50653
58 4 38 54872
58 5 75 421875
58 6 76 438976
58 7 77 456533
58 8 94 830584
58 9 95 857375
58 10 113 1442897
58 11 114 1481544
58 12 115 1520875
58 13 132 2299968
58 14 152 3511808
59 1 1 1
59 2 28 21952
59 3 29 24389
59 4 86 636056
59 5 88 681472
59 6 115 1520875
59 7 116 1560896
59 8 117 1601613
59 9 144 2985984
59 10 145 3048625
60 1 1 1
60 2 117 1601613
60 3 118 1643032
60 4 119 1685159
61 1 1 1
61 2 4 64
61 3 5 125
61 4 16 4096
61 5 20 8000
61 6 25 15625
61 7 31 29791
61 8 40 64000
61 9 64 262144
61 10 65 274625
61 11 69 328509
61 12 71 357911
61 13 79 493039
61 14 80 512000
61 15 81 531441
61 16 84 592704
61 17 85 614125
61 18 89 704969
61 19 91 753571
61 20 95 857375
61 21 99 970299
61 22 100 1000000
61 23 101 1030301
61 24 104 1124864
61 25 109 1295029
61 26 115 1520875
61 27 116 1560896
61 28 119 1685159
61 29 120 1728000
61 30 121 1771561
61 31 125 1953125
61 32 129 2146689
61 33 131 2248091
61 34 139 2685619
61 35 149 3307949
62 1 1 1
62 2 121 1771561
62 3 122 1815848
62 4 123 1860867
63 1 1 1
63 2 32 32768
63 3 93 804357
63 4 94 830584
63 5 123 1860867
63 6 124 1906624
63 7 125 1953125
64 1 1 1
64 2 8 512
64 3 28 21952
64 4 36 46656
64 5 71 357911
64 6 98 941192
64 7 125 1953125
64 8 126 2000376
64 9 127 2048383
64 10 154 3652264
64 11 161 4173281
65 1 1 1
65 2 31 29791
65 3 95 857375
65 4 127 2048383
65 5 128 2097152
65 6 129 2146689
66 1 1 1
66 2 26 17576
66 3 79 493039
66 4 90 729000
66 5 104 1124864
66 6 105 1157625
66 7 116 1560896
66 8 129 2146689
66 9 130 2197000
66 10 131 2248091
66 11 155 3723875
67 1 1 1
67 2 21 9261
67 3 32 32768
67 4 33 35937
67 5 45 91125
67 6 54 157464
67 7 55 166375
67 8 56 175616
67 9 87 658503
67 10 88 681472
67 11 89 704969
67 12 98 941192
67 13 100 1000000
67 14 109 1295029
67 15 110 1331000
67 16 111 1367631
67 17 120 1728000
67 18 121 1771561
67 19 131 2248091
67 20 132 2299968
67 21 133 2352637
67 22 142 2863288
67 23 143 2924207
67 24 144 2985984
67 25 153 3581577
67 26 164 4410944
67 27 165 4492125
67 28 175 5359375
67 29 231 12326391
68 1 1 1
68 2 133 2352637
68 3 134 2406104
68 4 135 2460375
69 1 1 1
69 2 17 4913
69 3 35 42875
69 4 101 1030301
69 5 103 1092727
69 6 119 1685159
69 7 120 1728000
69 8 135 2460375
69 9 136 2515456
69 10 137 2571353
69 11 152 3511808
69 12 153 3581577
69 13 169 4826809
70 1 1 1
70 2 22 10648
70 3 47 103823
70 4 91 753571
70 5 92 778688
70 6 93 804357
70 7 114 1481544
70 8 115 1520875
70 9 116 1560896
70 10 137 2571353
70 11 138 2628072
70 12 139 2685619
70 13 160 4096000
70 14 183 6128487
71 1 1 1
71 2 6 216
71 3 36 46656
71 4 49 117649
71 5 55 166375
71 6 64 262144
71 7 76 438976
71 8 90 729000
71 9 91 753571
71 10 99 970299
71 11 105 1157625
71 12 106 1191016
71 13 111 1367631
71 14 119 1685159
71 15 120 1728000
71 16 125 1953125
71 17 126 2000376
71 18 134 2406104
71 19 139 2685619
71 20 140 2744000
71 21 141 2803221
71 22 146 3112136
71 23 155 3723875
71 24 161 4173281
71 25 195 7414875
72 1 1 1
72 2 141 2803221
72 3 142 2863288
72 4 143 2924207
73 1 1 1
73 2 8 512
73 3 35 42875
73 4 45 91125
73 5 55 166375
73 6 64 262144
73 7 80 512000
73 8 81 531441
73 9 89 704969
73 10 99 970299
73 11 107 1225043
73 12 117 1601613
73 13 135 2460375
73 14 143 2924207
73 15 144 2985984
73 16 145 3048625
73 17 152 3511808
74 1 1 1
74 2 145 3048625
74 3 146 3112136
74 4 147 3176523
75 1 1 1
75 2 36 46656
75 3 37 50653
75 4 110 1331000
75 5 112 1404928
75 6 147 3176523
75 7 148 3241792
75 8 149 3307949
76 1 1 1
76 2 26 17576
76 3 49 117649
76 4 99 970299
76 5 100 1000000
76 6 101 1030301
76 7 126 2000376
76 8 149 3307949
76 9 150 3375000
76 10 151 3442951
76 11 176 5451776
77 1 1 1
77 2 19 6859
77 3 39 59319
77 4 57 185193
77 5 96 884736
77 6 113 1442897
77 7 115 1520875
77 8 132 2299968
77 9 133 2352637
77 10 151 3442951
77 11 152 3511808
77 12 153 3581577
77 13 171 5000211
77 14 189 6751269
77 15 208 8998912
77 16 209 9129329
78 1 1 1
78 2 43 79507
78 3 55 166375
78 4 99 970299
78 5 120 1728000
78 6 132 2299968
78 7 133 2352637
78 8 153 3581577
78 9 154 3652264
78 10 155 3723875
78 11 175 5359375
79 1 1 1
79 2 27 19683
79 3 40 64000
79 4 51 132651
79 5 64 262144
79 6 65 274625
79 7 66 287496
79 8 103 1092727
79 9 104 1124864
79 10 105 1157625
79 11 117 1601613
79 12 118 1643032
79 13 129 2146689
79 14 131 2248091
79 15 142 2863288
79 16 143 2924207
79 17 155 3723875
79 18 156 3796416
79 19 157 3869893
79 20 168 4741632
79 21 169 4826809
79 22 170 4913000
79 23 182 6028568
79 24 183 6128487
79 25 194 7301384
79 26 248 15252992
80 1 1 1
80 2 157 3869893
80 3 158 3944312
80 4 159 4019679
81 1 1 1
81 2 9 729
81 3 39 59319
81 4 55 166375
81 5 89 704969
81 6 105 1157625
81 7 111 1367631
81 8 119 1685159
81 9 145 3048625
81 10 151 3442951
81 11 159 4019679
81 12 160 4096000
81 13 161 4173281
81 14 176 5451776
81 15 215 9938375
82 1 1 1
82 2 161 4173281
82 3 162 4251528
82 4 163 4330747
83 1 1 1
83 2 40 64000
83 3 41 68921
83 4 122 1815848
83 5 124 1906624
83 6 163 4330747
83 7 164 4410944
83 8 165 4492125
84 1 1 1
84 2 165 4492125
84 3 166 4574296
84 4 167 4657463
85 1 1 1
85 2 7 343
85 3 8 512
85 4 20 8000
85 5 28 21952
85 6 43 79507
85 7 48 110592
85 8 49 117649
85 9 55 166375
85 10 56 175616
85 11 64 262144
85 12 91 753571
85 13 92 778688
85 14 97 912673
85 15 99 970299
85 16 105 1157625
85 17 111 1367631
85 18 112 1404928
85 19 113 1442897
85 20 119 1685159
85 21 120 1728000
85 22 125 1953125
85 23 127 2048383
85 24 132 2299968
85 25 133 2352637
85 26 139 2685619
85 27 140 2744000
85 28 141 2803221
85 29 147 3176523
85 30 148 3241792
85 31 153 3581577
85 32 160 4096000
85 33 161 4173281
85 34 167 4657463
85 35 168 4741632
85 36 169 4826809
85 37 175 5359375
85 38 176 5451776
85 39 181 5929741
85 40 183 6128487
85 41 188 6644672
85 42 189 6751269
85 43 203 8365427
85 44 209 9129329
86 1 1 1
86 2 34 39304
86 3 101 1030301
86 4 120 1728000
86 5 135 2460375
86 6 136 2515456
86 7 154 3652264
86 8 169 4826809
86 9 170 4913000
86 10 171 5000211
86 11 205 8615125
86 12 220 10648000
87 1 1 1
87 2 44 85184
87 3 129 2146689
87 4 130 2197000
87 5 171 5000211
87 6 172 5088448
87 7 173 5177717
87 8 214 9800344
88 1 1 1
88 2 29 24389
88 3 58 195112
88 4 115 1520875
88 5 116 1560896
88 6 117 1601613
88 7 144 2985984
88 8 145 3048625
88 9 146 3112136
88 10 173 5177717
88 11 174 5268024
88 12 175 5359375
89 1 1 1
89 2 21 9261
89 3 32 32768
89 4 43 79507
89 5 55 166375
89 6 56 175616
89 7 99 970299
89 8 120 1728000
89 9 121 1771561
89 10 131 2248091
89 11 144 2985984
89 12 175 5359375
89 13 176 5451776
89 14 177 5545233
89 15 209 9129329
90 1 1 1
90 2 177 5545233
90 3 178 5639752
90 4 179 5735339
91 1 1 1
91 2 9 729
91 3 26 17576
91 4 35 42875
91 5 44 85184
91 6 45 91125
91 7 54 157464
91 8 55 166375
91 9 71 357911
91 10 81 531441
91 11 99 970299
91 12 100 1000000
91 13 109 1295029
91 14 116 1560896
91 15 126 2000376
91 16 134 2406104
91 17 136 2515456
91 18 144 2985984
91 19 154 3652264
91 20 161 4173281
91 21 171 5000211
91 22 179 5735339
91 23 180 5832000
91 24 181 5929741
91 25 189 6751269
91 26 199 7880599
91 27 206 8741816
91 28 216 10077696
91 29 251 15813251
92 1 1 1
92 2 78 474552
92 3 181 5929741
92 4 182 6028568
92 5 183 6128487
92 6 196 7529536
92 7 209 9129329
93 1 1 1
93 2 47 103823
93 3 137 2571353
93 4 139 2685619
93 5 161 4173281
93 6 183 6128487
93 7 184 6229504
93 8 185 6331625
93 9 229 12008989
94 1 1 1
94 2 30 27000
94 3 63 250047
94 4 123 1860867
94 5 124 1906624
94 6 125 1953125
94 7 154 3652264
94 8 155 3723875
94 9 156 3796416
94 10 185 6331625
94 11 186 6434856
94 12 187 6539203
95 1 1 1
95 2 48 110592
95 3 141 2803221
95 4 142 2863288
95 5 187 6539203
95 6 188 6644672
95 7 189 6751269
95 8 234 12812904
96 1 1 1
96 2 56 175616
96 3 115 1520875
96 4 134 2406104
96 5 151 3442951
96 6 171 5000211
96 7 189 6751269
96 8 190 6859000
96 9 191 6967871
96 10 210 9261000
97 1 1 1
97 2 31 29791
97 3 47 103823
97 4 65 274625
97 5 111 1367631
97 6 113 1442897
97 7 127 2048383
97 8 128 2097152
97 9 129 2146689
97 10 143 2924207
97 11 159 4019679
97 12 160 4096000
97 13 161 4173281
97 14 175 5359375
97 15 191 6967871
97 16 192 7077888
97 17 193 7189057
97 18 207 8869743
97 19 239 13651919
98 1 1 1
98 2 193 7189057
98 3 194 7301384
98 4 195 7414875
99 1 1 1
99 2 48 110592
99 3 49 117649
99 4 146 3112136
99 5 148 3241792
99 6 195 7414875
99 7 196 7529536
99 8 197 7645373
100 1 1 1
100 2 10 1000
100 3 45 91125
100 4 109 1295029
100 5 143 2924207
100 6 153 3581577
100 7 154 3652264
100 8 188 6644672
100 9 197 7645373
100 10 198 7762392
100 11 199 7880599
100 12 208 8998912

Created by this little MUMPS program:

s NIL="" f B=2:1:100 s CC=0 f A=1:1 s (C,C0)=A**3,S=0 x "F D=0:1 q:'C  s S=C#B+S,C=C\B" s F=B-1*D<A q:F  i S=A s CC=CC+1 w B," ",CC," ",A," ",C0,!

The most remarkable thing about this, I think, is the observation that for a given base B, it’s easy to prove this list is complete, by noting that there can be no A greater than the number of digits in A3 times B-1. This is what the “set F=B-1*D<A quit:F” in my little program takes advantage of be able to stop searching for a given base B.

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