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Orderings of rational numbers
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Cantor ordering of positive rational numbers
The Cantor ordering of positive rational numbers first orders the positive rational numbers by increasing sum of numerators and denominators, then by increasing numerators, when the numerator is coprime to the denominator.
The balanced ternary digits {-1,0,+1} are represented here as {2,0,1} in the "Fermi-Dirac representation of positive rational numbers" below.
n num+den num den Fermi-Dirac factorization Fermi-Dirac representation A020652 A020653 1 2 1 / 1 Empty product 0 2 3 1 / 2 (1/2) 2 3 3 2 / 1 2 1 4 4 1 / 3 (1/3) 20 5 4 3 / 1 3 10 6 5 1 / 4 (1/8)*2 20001 7 5 2 / 3 (1/3)*2 21 8 5 3 / 2 3*(1/2) 12 9 5 4 / 1 8*(1/2) 10002 10 6 1 / 5 (1/5) 200 11 6 5 / 1 5 100 12 7 1 / 6 (1/3)*(1/2) 22 13 7 2 / 5 (1/5)*2 201 14 7 3 / 4 (1/8)*3*2 20011 15 7 4 / 3 8*(1/3)*(1/2) 10022 16 7 5 / 2 5*(1/2) 102 17 7 6 / 1 3*2 11 18 8 1 / 7 (1/7) 2000 19 8 3 / 5 (1/5)*3 210 20 8 5 / 3 5*(1/3) 120 21 8 7 / 1 7 1000 22 9 1 / 8 (1/8) 20000 23 9 2 / 7 (1/7)*2 2001 24 9 4 / 5 8*(1/5)*(1/2) 10202 25 9 5 / 4 (1/8)*5*2 20101 26 9 7 / 2 7*(1/2) 1002 27 9 8 / 1 8 10000 28 10 1 / 9 (1/27)*3 20000000010 29 10 3 / 7 (1/7)*3 2010 30 10 7 / 3 7*(1/3) 1020 31 10 9 / 1 27*(1/3) 10000000020 32 11 1 / 10 (1/5)*(1/2) 202 33 11 2 / 9 (1/27)*3*2 20000000011 34 11 3 / 8 (1/8)*3 20010 35 11 4 / 7 8*(1/7)*(1/2) 12002 36 11 5 / 6 5*(1/3)*(1/2) 122 37 11 6 / 5 (1/5)*3*2 211 38 11 7 / 4 (1/8)*7*2 21001 39 11 8 / 3 8*(1/3) 10020 40 11 9 / 2 27*(1/3)*(1/2) 10000000022 41 11 10 / 1 5*2 101 42 12 1 / 11 (1/11) 200000 43 12 5 / 7 (1/7)*5 2100 44 12 7 / 5 7*(1/5) 1200 45 12 11 / 1 11 100000 46 13 1 / 12 (1/8)*(1/3)*2 20021 47 13 2 / 11 (1/11)*2 200001 48 13 3 / 10 (1/5)*3*(1/2) 212 49 13 4 / 9 (1/27)*8*3*(1/2) 20000010012 50 13 5 / 8 (1/8)*5 20100 51 13 6 / 7 (1/7)*3*2 2011 52 13 7 / 6 7*(1/3)*(1/2) 1022 53 13 8 / 5 8*(1/5) 10200 54 13 9 / 4 27*(1/8)*(1/3)*2 10000020021 55 13 10 / 3 5*(1/3)*2 121 56 13 11 / 2 11*(1/2) 100002 57 13 12 / 1 8*3*(1/2) 10012 58 14 1 / 13 59 14 3 / 11 60 14 5 / 9 61 14 9 / 5 62 14 11 / 3 63 14 13 / 1 64 15 1 / 14 65 15 2 / 13 66 15 4 / 11 67 15 7 / 8 68 15 8 / 7 69 15 11 / 4 70 15 13 / 2 71 15 14 / 1 72 16 1 / 15 73 16 3 / 13 74 16 5 / 11 75 16 7 / 9 76 16 9 / 7 77 16 11 / 5 78 16 13 / 3 79 16 15 / 1 80 17 1 / 16 81 17 2 / 15 82 17 3 / 14 83 17 4 / 13 84 17 5 / 12 85 17 6 / 11 86 17 7 / 10 87 17 8 / 9 88 17 9 / 8 89 17 10 / 7 90 17 11 / 6 91 17 12 / 5 92 17 13 / 4 93 17 14 / 3 94 17 15 / 2 95 17 16 / 1 96 18 1 / 17 97 18 5 / 13 98 18 7 / 11 99 18 11 / 7 100 18 13 / 5 101 18 17 / 1 102 19 1 / 18 103 19 2 / 17 104 19 3 / 16 105 19 4 / 15 106 19 5 / 14 107 19 6 / 13 108 19 7 / 12 109 19 8 / 11 110 19 9 / 10 111 19 10 / 9 112 19 11 / 8 113 19 12 / 7 114 19 13 / 6 115 19 14 / 5 116 19 15 / 4 117 19 16 / 3 118 19 17 / 2 119 19 18 / 1 120 20 1 / 19 121 20 3 / 17 122 20 7 / 13 123 20 9 / 11 124 20 11 / 9 125 20 13 / 7 126 20 17 / 3 127 20 19 / 1 128 21 1 / 20 129 21 2 / 19 130 21 4 / 17 131 21 5 / 16 132 21 8 / 13 133 21 10 / 11 134 21 11 / 10 135 21 13 / 8 136 21 16 / 5 137 21 17 / 4 138 21 19 / 2 139 21 20 / 1 140 22 1 / 21 141 22 3 / 19 142 22 5 / 17 143 22 7 / 15 144 22 9 / 13 145 22 13 / 9 146 22 15 / 7 147 22 17 / 5 148 22 19 / 3 149 22 21 / 1 150 23 1 / 22 151 23 2 / 21 152 23 3 / 20 153 23 4 / 19 154 23 5 / 18 155 23 6 / 17 156 23 7 / 16 157 23 8 / 15 158 23 9 / 14 159 23 10 / 13 160 23 11 / 12 161 23 12 / 11 162 23 13 / 10 163 23 14 / 9 164 23 15 / 8 165 23 16 / 7 166 23 17 / 6 167 23 18 / 5 168 23 19 / 4 169 23 20 / 3 170 23 21 / 2 171 23 22 / 1 172 24 1 / 23 173 24 5 / 19 174 24 7 / 17 175 24 11 / 13 176 24 13 / 11 177 24 17 / 7 178 24 19 / 5 179 24 23 / 1 180 25 1 / 24 181 25 2 / 23 182 25 3 / 22 183 25 4 / 21 184 25 6 / 19 185 25 7 / 18 186 25 8 / 17 187 25 9 / 16 188 25 11 / 14 189 25 12 / 13 190 25 13 / 12 191 25 14 / 11 192 25 16 / 9 193 25 17 / 8 194 25 18 / 7 195 25 19 / 6 196 25 21 / 4 197 25 22 / 3 198 25 23 / 2 199 25 24 / 1 200 26 1 / 25 (1/125)*5 2...(29 0's)...100 201 26 3 / 23 202 26 5 / 21 203 26 7 / 19 204 26 9 / 17 205 26 11 / 15 206 26 15 / 11 207 26 17 / 9 208 26 19 / 7 209 26 21 / 5 210 26 23 / 3 211 26 25 / 1 125*(1/5) 1...(29 0's)...200
Ordering of positive rational numbers by increasing representation based on their factorization into prime powers with powers of three as exponents
The factorization of positive rational numbers into prime powers of the form p^(3^k), k >= 0, A186285) and their multiplicative inverses, allows each of those prime powers and their multiplicative inverses to be used at most once, since this corresponds to the balanced ternary representation of the exponents of the prime powers and their multiplicative inverses of the "Bose-Einstein factorization of positive rational numbers", i.e. the classic prime factorization of positive rational numbers. (Cf. A050376 comments) This is sometimes called the "Fermi-Dirac factorization of positive rational numbers" by analogy with the Fermi-Dirac distribution.
According to the Bose-Einstein distribution of particles, an unlimited number of particles may occupy the same state. On the other hand, according to the Fermi-Dirac distribution, no two particles can occupy the same state (the Pauli Exclusion Principle). Unique factorizations of the positive rational numbers (in reduced form) by primes (A000040) and their multiplicative inverses, and over terms of A186285 and their multiplicative inverses, one can compare with these two distributions in physics of particles. In the correspondence with this, the factorizations over primes and their multiplicative inverses one can call "Bose-Einstein factorization of positive rational numbers", while the factorizations over distinct terms of A186285 and their multiplicative inverses one can call "Fermi-Dirac factorization of positive rational numbers". (Cf. A050376 comments)
The numbers of the form p^(3^k), where p is prime and k >= 0, might thus be called the "Fermi-Dirac primes of the positive rational numbers", while the "Bose-Einstein primes of the positive rational numbers"]] (which are the same as the "Bose-Einstein primes of the positive integers") are the classic primes.
The "Fermi-Dirac primes of the positive integers", then are prime powers of the form p^(2^k), k >= 0. (A050376)
Here is a table of the ordering of positive rational numbers by balanced ternary representation of the "factorization" into terms of A186285 (prime powers with a power of three as exponent.)
In the following table the balanced ternary digits {-1, 0, +1} are represented as {2, 0, 1}, 2 being congruent to -1 modulo 3. Rnum and Rden are the reduced numerators and denominators, i.e. divided by GCD(Num, Den).
Representation | 127 | 125 | 113 | 109 | 107 | 103 | 101 | 97 | 89 | 83 | 79 | 73 | 71 | 67 | 61 | 59 | 53 | 47 | 43 | 41 | 37 | 31 | 29 | 27 | 23 | 19 | 17 | 13 | 11 | 8 | 7 | 5 | 3 | 2 | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | 0 | |||||||||||||||||||||||||||||||||||
1 | 1 | 1 | ||||||||||||||||||||||||||||||||||
2 | 12 | 1 | -1 | |||||||||||||||||||||||||||||||||
3 | 10 | 1 | 0 | |||||||||||||||||||||||||||||||||
4 | 11 | 1 | 1 | |||||||||||||||||||||||||||||||||
5 | 122 | 1 | -1 | -1 | ||||||||||||||||||||||||||||||||
6 | 120 | 1 | -1 | 0 | ||||||||||||||||||||||||||||||||
7 | 121 | 1 | -1 | 1 | ||||||||||||||||||||||||||||||||
8 | 102 | 1 | 0 | -1 | ||||||||||||||||||||||||||||||||
9 | 100 | 1 | 0 | 0 |
Bal Num / Den GCD Rnum / Rden Tern 1 -1 0 1 / 1 1 1 / 1 1 2 / 1 1 2 / 1 12 3 / 2 1 3 / 2 10 3 / 1 1 3 / 1 11 6 / 1 1 6 / 1 122 5 / 6 1 5 / 6 120 5 / 3 1 5 / 3 121 10 / 3 1 10 / 3 102 5 / 2 1 5 / 2 100 5 / 1 1 5 / 1 101 10 / 1 1 10 / 1 112 15 / 2 1 15 / 2 110 15 / 1 1 15 / 1 111 30 / 1 1 30 / 1 1222 7 / 30 1 7 / 30 1220 7 / 15 1 7 / 15 1221 14 / 15 1 14 / 15 1202 7 / 10 1 7 / 10 1200 7 / 5 1 7 / 5 1201 14 / 5 1 14 / 5 1212 21 / 10 1 21 / 10 1210 21 / 5 1 21 / 5 1211 42 / 5 1 42 / 5 1022 7 / 6 1 7 / 6 1020 7 / 3 1 7 / 3 1021 14 / 3 1 14 / 3 1002 7 / 2 1 7 / 2 1000 7 / 1 1 7 / 1 1001 14 / 1 1 14 / 1 1012 21 / 2 1 21 / 2 1010 21 / 1 1 21 / 1 1011 42 / 1 1 42 / 1 1122 35 / 6 1 35 / 6 1120 35 / 3 1 35 / 3 1121 70 / 3 1 70 / 3 1102 35 / 2 1 35 / 2 1100 35 / 1 1 35 / 1 1101 70 / 1 1 70 / 1 1112 105 / 2 1 105 / 2 1110 105 / 1 1 105 / 1 1111 210 / 1 1 210 / 1 12222 8 / 210 2 4 / 105 12220 8 / 105 1 8 / 105 12221 16 / 105 1 16 / 105 12202 8 / 70 2 4 / 35 12200 8 / 35 1 8 / 35 12201 16 / 35 1 16 / 35 12212 24 / 70 2 12 / 35 12210 24 / 35 1 24 / 35 12211 48 / 35 1 48 / 35 12022 8 / 42 2 4 / 21 12020 8 / 21 1 8 / 21 12021 16 / 21 1 16 / 21 12002 8 / 14 2 4 / 7 12000 8 / 7 1 8 / 7 12001 16 / 7 1 16 / 7 12012 24 / 14 2 12 / 7 12010 24 / 7 1 24 / 7 12011 48 / 7 1 48 / 7 12122 40 / 42 2 20 / 21 12120 40 / 21 1 40 / 21 12121 80 / 21 1 80 / 21 12102 40 / 14 2 20 / 7 12100 40 / 7 1 40 / 7 12101 80 / 7 1 80 / 7 12112 120 / 14 2 60 / 7 12110 120 / 7 1 120 / 7 12111 240 / 7 1 240 / 7 10222 8 / 30 2 4 / 15 10220 8 / 15 1 8 / 15 10221 16 / 15 1 16 / 15 10202 8 / 10 2 4 / 5 10200 8 / 5 1 8 / 5 10201 16 / 5 1 16 / 5 10212 24 / 10 2 12 / 5 10210 24 / 5 1 24 / 5 10211 48 / 5 1 48 / 5 10022 8 / 6 2 4 / 3 10020 8 / 3 1 8 / 3 10021 16 / 3 1 16 / 3 10002 8 / 2 2 4 / 1 10000 8 / 1 1 8 / 1 10001 16 / 1 1 16 / 1 10012 24 / 2 2 12 / 1 10010 24 / 1 1 24 / 1 10011 48 / 1 1 48 / 1 10122 40 / 6 2 20 / 3 10120 40 / 3 1 40 / 3 10121 80 / 3 1 80 / 3 10102 40 / 2 2 20 / 1 10100 40 / 1 1 40 / 1 10101 80 / 1 1 80 / 1 10112 120 / 2 2 60 / 1 10110 120 / 1 1 120 / 1 10111 240 / 1 1 240 / 1 11222 56 / 30 2 28 / 15 11220 56 / 15 1 56 / 15 11221 112 / 15 1 112 / 15 11202 56 / 10 2 28 / 5 11200 56 / 5 1 56 / 5 11201 112 / 5 1 112 / 5 11212 168 / 10 2 84 / 5 11210 168 / 5 1 168 / 5 11211 336 / 5 1 336 / 5 11022 56 / 6 2 28 / 3 11020 56 / 3 1 56 / 3 11021 112 / 3 1 112 / 3 11002 56 / 2 2 28 / 1 11000 56 / 1 1 56 / 1 11001 112 / 1 1 112 / 1 11012 168 / 2 2 84 / 1 11010 168 / 1 1 168 / 1 11011 336 / 1 1 336 / 1 11122 280 / 6 2 140 / 3 11120 280 / 3 1 280 / 3 11121 560 / 3 1 560 / 3 11102 280 / 2 2 140 / 1 11100 280 / 1 1 280 / 1 11101 560 / 1 1 560 / 1 11112 840 / 2 2 420 / 1 11110 840 / 1 1 840 / 1 11111 1680 / 1 1 1680 / 1 122222 11 / 1680 1 11 / 1680 122220 11 / 840 1 11 / 840 122221 22 / 840 2 11 / 420 122202 11 / 560 1 11 / 560 122200 11 / 280 1 11 / 280 122201 22 / 280 2 11 / 140 122212 33 / 560 1 33 / 560 122210 33 / 280 1 33 / 280 122211 66 / 280 2 33 / 140 122022 11 / 336 1 11 / 336 122020 11 / 168 1 11 / 168 122021 22 / 168 2 11 / 84 122002 11 / 112 1 11 / 112 122000 11 / 56 1 11 / 56 122001 22 / 56 2 11 / 28 122012 33 / 112 1 33 / 112 122010 33 / 56 1 33 / 56 122011 66 / 56 2 33 / 28 122122 55 / 336 1 55 / 336 122120 55 / 168 1 55 / 168 122121 110 / 168 2 55 / 84 122102 55 / 112 1 55 / 112 122100 55 / 56 1 55 / 56 122101 110 / 56 2 55 / 28 122112 165 / 112 1 165 / 112 122110 165 / 56 1 165 / 56 122111 330 / 56 2 165 / 28 120222 11 / 240 1 11 / 240 120220 11 / 120 1 11 / 120 120221 22 / 120 2 11 / 60 120202 11 / 80 1 11 / 80 120200 11 / 40 1 11 / 40 120201 22 / 40 2 11 / 20 120212 33 / 80 1 33 / 80 120210 33 / 40 1 33 / 40 120211 66 / 40 2 33 / 20 120022 11 / 48 1 11 / 48 120020 11 / 24 1 11 / 24 120021 22 / 24 2 11 / 12 120002 11 / 16 1 11 / 16 120000 11 / 8 1 11 / 8 120001 22 / 8 2 11 / 4 120012 33 / 16 1 33 / 16 120010 33 / 8 1 33 / 8 120011 66 / 8 2 33 / 4 120122 55 / 48 1 55 / 48 120120 55 / 24 1 55 / 24 120121 110 / 24 2 55 / 12 120102 55 / 16 1 55 / 16 120100 55 / 8 1 55 / 8 120101 110 / 8 2 55 / 4 120112 165 / 16 1 165 / 16 120110 165 / 8 1 165 / 8 120111 330 / 8 2 165 / 4 121222 77 / 240 1 77 / 240 121220 77 / 120 1 77 / 120 121221 154 / 120 2 77 / 60 121202 77 / 80 1 77 / 80 121200 77 / 40 1 77 / 40 121201 154 / 40 2 77 / 20 121212 231 / 80 1 231 / 80 121210 231 / 40 1 231 / 40 121211 462 / 40 2 231 / 20 121022 77 / 48 1 77 / 48 121020 77 / 24 1 77 / 24 121021 154 / 24 2 77 / 12 121002 77 / 16 1 77 / 16 121000 77 / 8 1 77 / 8 121001 154 / 8 2 77 / 4 121012 231 / 16 1 231 / 16 121010 231 / 8 1 231 / 8 121011 462 / 8 2 231 / 4 121122 385 / 48 1 385 / 48 121120 385 / 24 1 385 / 24 121121 770 / 24 2 385 / 12 121102 385 / 16 1 385 / 16 121100 385 / 8 1 385 / 8 121101 770 / 8 2 385 / 4 121112 1155 / 16 1 1155 / 16 121110 1155 / 8 1 1155 / 8 121111 2310 / 8 2 1155 / 4 102222 11 / 210 1 11 / 210 102220 11 / 105 1 11 / 105 102221 22 / 105 1 22 / 105 102202 11 / 70 1 11 / 70 102200 11 / 35 1 11 / 35 102201 22 / 35 1 22 / 35 102212 33 / 70 1 33 / 70 102210 33 / 35 1 33 / 35 102211 66 / 35 1 66 / 35 102022 11 / 42 1 11 / 42 102020 11 / 21 1 11 / 21 102021 22 / 21 1 22 / 21 102002 11 / 14 1 11 / 14 102000 11 / 7 1 11 / 7 102001 22 / 7 1 22 / 7 102012 33 / 14 1 33 / 14 102010 33 / 7 1 33 / 7 102011 66 / 7 1 66 / 7 102122 55 / 42 1 55 / 42 102120 55 / 21 1 55 / 21 102121 110 / 21 1 110 / 21 102102 55 / 14 1 55 / 14 102100 55 / 7 1 55 / 7 102101 110 / 7 1 110 / 7 102112 165 / 14 1 165 / 14 102110 165 / 7 1 165 / 7 102111 330 / 7 1 330 / 7 100222 11 / 30 1 11 / 30 100220 11 / 15 1 11 / 15 100221 22 / 15 1 22 / 15 100202 11 / 10 1 11 / 10 100200 11 / 5 1 11 / 5 100201 22 / 5 1 22 / 5 100212 33 / 10 1 33 / 10 100210 33 / 5 1 33 / 5 100211 66 / 5 1 66 / 5 100022 11 / 6 1 11 / 6 100020 11 / 3 1 11 / 3 100021 22 / 3 1 22 / 3 100002 11 / 2 1 11 / 2 100000 11 / 1 1 11 / 1 100001 22 / 1 1 22 / 1 100012 33 / 2 1 33 / 2 100010 33 / 1 1 33 / 1 100011 66 / 1 1 66 / 1 100122 55 / 6 1 55 / 6 100120 55 / 3 1 55 / 3 100121 110 / 3 1 110 / 3 100102 55 / 2 1 55 / 2 100100 55 / 1 1 55 / 1 100101 110 / 1 1 110 / 1 100112 165 / 2 1 165 / 2 100110 165 / 1 1 165 / 1 100111 330 / 1 1 330 / 1 101222 77 / 30 1 77 / 30 101220 77 / 15 1 77 / 15 101221 154 / 15 1 154 / 15 101202 77 / 10 1 77 / 10 101200 77 / 5 1 77 / 5 101201 154 / 5 1 154 / 5 101212 231 / 10 1 231 / 10 101210 231 / 5 1 231 / 5 101211 462 / 5 1 462 / 5 101022 77 / 6 1 77 / 6 101020 77 / 3 1 77 / 3 101021 154 / 3 1 154 / 3 101002 77 / 2 1 77 / 2 101000 77 / 1 1 77 / 1 101001 154 / 1 1 154 / 1 101012 231 / 2 1 231 / 2 101010 231 / 1 1 231 / 1 101011 462 / 1 1 462 / 1 101122 385 / 6 1 385 / 6 101120 385 / 3 1 385 / 3 101121 770 / 3 1 770 / 3 101102 385 / 2 1 385 / 2 101100 385 / 1 1 385 / 1 101101 770 / 1 1 770 / 1 101112 1155 / 2 1 1155 / 2 101110 1155 / 1 1 1155 / 1 101111 2310 / 1 1 2310 / 1 112222 88 / 210 2 44 / 105 112220 88 / 105 1 88 / 105 112221 176 / 105 1 176 / 105 112202 88 / 70 2 44 / 35 112200 88 / 35 1 88 / 35 112201 176 / 35 1 176 / 35 112212 264 / 70 2 132 / 35 112210 264 / 35 1 264 / 35 112211 528 / 35 1 528 / 35 112022 88 / 42 2 44 / 21 112020 88 / 21 1 88 / 21 112021 176 / 21 1 176 / 21 112002 88 / 14 2 44 / 7 112000 88 / 7 1 88 / 7 112001 176 / 7 1 176 / 7 112012 264 / 14 2 132 / 7 112010 264 / 7 1 264 / 7 112011 528 / 7 1 528 / 7 112122 440 / 42 2 220 / 21 112120 440 / 21 1 440 / 21 112121 880 / 21 1 880 / 21 112102 440 / 14 2 220 / 7 112100 440 / 7 1 440 / 7 112101 880 / 7 1 880 / 7 112112 1320 / 14 2 660 / 7 112110 1320 / 7 1 1320 / 7 112111 2640 / 7 1 2640 / 7 110222 88 / 30 2 44 / 15 110220 88 / 15 1 88 / 15 110221 176 / 15 1 176 / 15 110202 88 / 10 2 44 / 5 110200 88 / 5 1 88 / 5 110201 176 / 5 1 176 / 5 110212 264 / 10 2 132 / 5 110210 264 / 5 1 264 / 5 110211 528 / 5 1 528 / 5 110022 88 / 6 2 44 / 3 110020 88 / 3 1 88 / 3 110021 176 / 3 1 176 / 3 110002 88 / 2 2 44 / 1 110000 88 / 1 1 88 / 1 110001 176 / 1 1 176 / 1 110012 264 / 2 2 132 / 1 110010 264 / 1 1 264 / 1 110011 528 / 1 1 528 / 1 110122 440 / 6 2 220 / 3 110120 440 / 3 1 440 / 3 110121 880 / 3 1 880 / 3 110102 440 / 2 2 220 / 1 110100 440 / 1 1 440 / 1 110101 880 / 1 1 880 / 1 110112 1320 / 2 2 660 / 1 110110 1320 / 1 1 1320 / 1 110111 2640 / 1 1 2640 / 1 111222 616 / 30 2 308 / 15 111220 616 / 15 1 616 / 15 111221 1232 / 15 1 1232 / 15 111202 616 / 10 2 308 / 5 111200 616 / 5 1 616 / 5 111201 1232 / 5 1 1232 / 5 111212 1848 / 10 2 924 / 5 111210 1848 / 5 1 1848 / 5 111211 3696 / 5 1 3696 / 5 111022 616 / 6 2 308 / 3 111020 616 / 3 1 616 / 3 111021 1232 / 3 1 1232 / 3 111002 616 / 2 2 308 / 1 111000 616 / 1 1 616 / 1 111001 1232 / 1 1 1232 / 1 111012 1848 / 2 2 924 / 1 111010 1848 / 1 1 1848 / 1 111011 3696 / 1 1 3696 / 1 111122 3080 / 6 2 1540 / 3 111120 3080 / 3 1 3080 / 3 111121 6160 / 3 1 6160 / 3 111102 3080 / 2 2 1540 / 1 111100 3080 / 1 1 3080 / 1 111101 6160 / 1 1 6160 / 1 111112 9240 / 2 2 4620 / 1 111110 9240 / 1 1 9240 / 1 111111 18480 / 1 1 18480 / 1
See also
- Orderings of rational numbers and permutation of the rational numbers
- Orderings of algebraic numbers and permutation of the algebraic numbers
- Orderings of integers and permutation of the integers
- Orderings of positive integers and permutation of the positive integers
External links
- David M. Bradley, "Counting the Positive Rationals: A Brief Survey," 2005.
- Kevin McCrimmon, "Enumeration of the Positive Rationals," The American Mathematical Monthly, Vol. 67, No. 9, Nov., 1960.
- Gerald Freilich, "A Denumerability Formula for the Rationals," The American Mathematical Monthly, Vol. 72, No. 9, Nov., 1965.