Rade Posted June 15, 2013 Report Posted June 15, 2013 (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 June 17, 2013 by Rade Quote
phision Posted June 15, 2013 Report Posted June 15, 2013 (edited) 8,18 & 27 are not Prime numbers! See: https://en.wikipedia...ki/Prime_number Edited June 15, 2013 by phision Quote
Rade Posted June 15, 2013 Author Report Posted June 15, 2013 (edited) 8,18 & 27 are not Prime numbers! See: https://en.wikipedia...ki/Prime_numberOK, thanks...yet another error on my part. I will make edits on OP. But then, why would 17 which is a prime number show the above pattern and 18 which is not prime also show the cubing pattern ? == Edited June 15, 2013 by Rade Quote
phillip1882 Posted June 17, 2013 Report Posted June 17, 2013 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. Turtle 1 Quote
Turtle Posted June 17, 2013 Report Posted June 17, 2013 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. It may be the case that these sequences have been discovered, but I'm unsure how to search such a thing. Quote
Rade Posted June 17, 2013 Author Report Posted June 17, 2013 (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 June 17, 2013 by Rade Quote
CraigD Posted June 17, 2013 Report Posted June 17, 2013 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. Turtle and Rade 2 Quote
Rade Posted June 17, 2013 Author Report Posted June 17, 2013 Thank you CraigD for your time to reply and excellent table, and help with thread changes. Quote
Recommended Posts
Join the conversation
You can post now and register later. If you have an account, sign in now to post with your account.