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Orderings of ordered prime signatures
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Contents
- 1 Binary representation ordering of the ordered prime signatures
- 2 Increasing least integers of ordered prime signatures graded by prime signatures ordered by increasing least integers of prime signatures
- 3 Ordered prime signatures in graded colexicographic order
- 4 Ordered prime signatures in the order of increasing smallest numbers of ordered prime signatures
- 5 Sequences
- 6 See also
- 7 Notes
Binary representation ordering of the ordered prime signatures
| Ordered prime signature | Numbers
{Distinct prime factors} |
OEIS
number |
Prime signature | ||
|---|---|---|---|---|---|
| 0 | 0[1] | { } | {1}
{{ }} |
{ } | |
| 1 | 1 | {1} | {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, ...}
{{2}, {3}, {5}, {7}, {11}, {13}, {17}, {19}, {23}, {29}, {31}, {37}, {41}, {43}, {47}, {53}, {59}, {61}, {67}, {71}, {73}, {79}, {83}, {89}, ...} |
A000040 | { } |
| 2 | 10 | {1,1} | {6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, ...}
{{2,3}, {2,5}, {2,7}, {3,5}, {3,7}, {2,11}, {2,13}, {3,11}, {2,17}, {5,7}, {2,19}, {3,13}, {2,23}, {3,17}, {5,11}, {3,19}, {2,29}, {2,31}, {5,13}, ...} |
A006881 | { } |
| 3 | 11 | {2} | {4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, ...}
{{2}, {3}, {5}, {7}, {11}, {13}, {17}, {19}, {23}, {29}, {31}, {37}, {41}, {43}, {47}, {53}, {59}, {61}, {67}, {71}, {73}, {79}, {83}, {89}, ...} |
A001248 | { } |
| 4 | 100 | {1,2} | {18, 50, 75, 98, 147, 242, 245, 338, 363, 507, 578, 605, 722, 845, 847, 867, 1058, 1083, 1183, 1445, 1587, 1682, 1805, ...}
{{2,3}, {2,5}, {3,5}, {2,7}, {3,7}, {2,11}, {5,7}, {2,13}, {3,11}, {3,13}, {2,17}, {5,11}, {2,19}, {5,13}, {7,11}, {3,17}, {2,23}, {3,19}, {7,13}, ...} |
A095990 | { } |
| 5 | 101 | {1,1,1} | {30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, 230, 231, 238, 246, 255, 258, ...}
{{2,3,5}, {2,3,7}, {2,3,11}, {2,5,7}, {2,3,13}, {2,3,17}, {3,5,7}, {2,5,11}, {2,3,19}, {2,5,13}, {2,3,23}, {2,7,11}, {3,5,11}, {2,5,17}, ...} |
A007304 | { } |
| 6 | 110 | {2,1} | {12, 20, 28, 44, 45, 52, 63, 68, 76, 92, 99, 116, 117, 124, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 244, 261, 268, ...}
{{2,3}, {2,5}, {2,7}, {2,11}, {3,5}, {2,13}, {3,7}, {2,17}, {2,19}, {2,23}, {3,11}, {2,29}, {3,13}, {2,31}, {2,37}, {3,17}, {2,41}, {3,19}, ...} |
A096156 | { } |
| 7 | 111 | {3} | {8, 27, 125, 343, 1331, 2197, 4913, 6859, 12167, 24389, 29791, 50653, 68921, 79507, 103823, 148877, 205379, 226981, ...}
{{2}, {3}, {5}, {7}, {11}, {13}, {17}, {19}, {23}, {29}, {31}, {37}, {41}, {43}, {47}, {53}, {59}, {61}, {67}, {71}, {73}, {79}, {83}, {89}, ...} |
A030078 | { } |
| 8 | 1000 | {1,3} | {54, 250, 375, 686, ...}
{{2,3}, {2,5}, {3,5}, {2,7}, ...} |
A?????? | { } |
| 9 | 1001 | {1,2,1} | {90, ...} | A?????? | { } |
| 10 | 1010 | {1,1,1,1} | {210, 330, 390, 462, 510, 546, 570, 690, 714, 770, 798, 858, 870, 910, 930, 966, 1110, 1122, ...} | A046386 | { } |
| 11 | 1011 | {1,1,2} | {150, ...} | A?????? | { } |
| 12 | 1100 | {2,2} | {36, 100, 196, 225, 441, 484, 676, 1089, 1156, 1225, 1444, 1521, 2116, 2601, 3025, 3249, ...} | A085986 | { } |
| 13 | 1101 | {2,1,1} | {60, ...} | A?????? | { } |
| 14 | 1110 | {3,1} | {24, 40, 56, 88, 104, 135, 136, ...}
{{2,3}, {2,5}, {2,7}, {2,11}, {2,13}, {3,5}, {2,17}, ...} |
A?????? | { } |
| 15 | 1111 | {4} | {16, 81, 625, 2401, 14641, 28561, 83521, 130321, 279841, 707281, 923521, 1874161, ...}
{{2}, {3}, {5}, {7}, {11}, {13}, {17}, {19}, {23}, {29}, {31}, {37}, {41}, {43}, {47}, {53}, {59}, {61}, {67}, {71}, {73}, {79}, {83}, {89}, ...} |
A030514 | { } |
| 16 | 10000 | {1,4} | {162, ...} | A?????? | { } |
| 17 | 10001 | {1,3,1} | {270, ...} | A?????? | { } |
| 18 | 10010 | {1,2,1,1} | {630, ...} | A?????? | { } |
| 19 | 10011 | {1,2,2} | {450 , ...} | A?????? | { } |
| 20 | 10100 | {1,1,1,2} | {1470, ...} | A?????? | { } |
| 21 | 10101 | {1,1,1,1,1} | {2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6270, 6510, 6630, 7410, 7590, ...} | A046387 | { } |
| 22 | 10110 | {1,1,2,1} | {1050, ...} | A?????? | { } |
| 23 | 10111 | {1,1,3} | {750, ...} | A?????? | { } |
| 24 | 11000 | {2,3} | {108, ...} | A?????? | { } |
| 25 | 11001 | {2,2,1} | {180, ...} | A?????? | { } |
| 26 | 11010 | {2,1,1,1} | {420, ...} | A?????? | { } |
| 27 | 11011 | {2,1,2} | {300, ...} | A?????? | { } |
| 28 | 11100 | {3,2} | {72, ...} | A?????? | { } |
| 29 | 11101 | {3,1,1} | {120, ...} | A?????? | { } |
| 30 | 11110 | {4,1} | {48, ...} | A?????? | { } |
| 31 | 11111 | {5} | {32, 243, 3125, 16807, 161051, 371293, 1419857, 2476099, 6436343, 20511149, 28629151, ...}
{{2}, {3}, {5}, {7}, {11}, {13}, {17}, {19}, {23}, {29}, {31}, {37}, {41}, {43}, {47}, {53}, {59}, {61}, {67}, {71}, {73}, {79}, {83}, {89}, ...} |
A050997 | { } |
| 32 | 100000 | {1,5} | {486, ...} | A?????? | { } |
| 33 | 100001 | {1,4,1} | {810, ...} | A?????? | { } |
| 34 | 100010 | {1,3,1,1} | {1890, ...} | A?????? | { } |
| 35 | 100011 | {1,3,2} | {1500, ...} | A?????? | { } |
| 36 | 100100 | {1,2,1,2} | {4410, ...} | A?????? | { } |
| 37 | 100101 | {1,2,1,1,1} | {6930, ...} | A?????? | { } |
| 38 | 100110 | {1,2,2,1} | {3150, ...} | A?????? | { } |
| 39 | 100111 | {1,2,3} | {2250, ...} | A?????? | { } |
| 40 | 101000 | {1,1,1,3} | {10290, ...} | A?????? | { } |
| 41 | 101001 | {1,1,1,2,1} | {16170, ...} | A?????? | { } |
| 42 | 101010 | {1,1,1,1,1,1} | {30030, ...} | A?????? | { } |
| 43 | 101011 | {1,1,1,1,2} | {25410, ...} | A?????? | { } |
| 44 | 101100 | {1,1,2,2} | {7350, ...} | A?????? | { } |
| 45 | 101101 | {1,1,2,1,1} | {11550, ...} | A?????? | { } |
| 46 | 101110 | {1,1,3,1} | {5250, ...} | A?????? | { } |
| 47 | 101111 | {1,1,4} | {3750, ...} | A?????? | { } |
| 48 | 110000 | {2,4} | {324 , ...} | A?????? | { } |
| 49 | 110001 | {2,3,1} | {540, ...} | A?????? | { } |
| 50 | 110010 | {2,2,1,1} | {1260, ...} | A?????? | { } |
| 51 | 110011 | {2,2,2} | {900, 1764, 4356, 4900, 6084, 10404, 11025, 12100, 12996, 16900, 19044, 23716, 27225, ...} | A162143 | { } |
| 52 | 110100 | {2,1,1,2} | {2940, ...} | A?????? | { } |
| 53 | 110101 | {2,1,1,1,1} | {4620, ...} | A?????? | { } |
| 54 | 110110 | {2,1,2,1} | {2100, ...} | A?????? | { } |
| 55 | 110111 | {2,1,3} | {1350, ...} | A?????? | { } |
| 56 | 111000 | {3,3} | {216, 1000, 2744, 3375, 9261, 10648, 17576, 35937, 39304, 42875, 54872, 59319, 97336, ...} | A162142 | { } |
| 57 | 111001 | {3,2,1} | {360, ...} | A?????? | { } |
| 58 | 111010 | {3,1,1,1} | {840, ...} | A?????? | { } |
| 59 | 111011 | {3,1,2} | {600, ...} | A?????? | { } |
| 60 | 111100 | {4,2} | {144, ...} | A?????? | { } |
| 61 | 111101 | {4,1,1} | {240, ...} | A?????? | { } |
| 62 | 111110 | {5,1} | {96, ...} | A?????? | { } |
| 63 | 111111 | {6} | {64, 729, 15625, 117649, 1771561, 4826809, 24137569, 47045881, 148035889, 594823321, ...}
{{2}, {3}, {5}, {7}, {11}, {13}, {17}, {19}, {23}, {29}, {31}, {37}, {41}, {43}, {47}, {53}, {59}, {61}, {67}, {71}, {73}, {79}, {83}, {89}, ...} |
A030516 | { } |
| 64 | 1000000 | {1,6} | {1458, ...} | A?????? | { } |
| 65 | 1000001 | {1,5,1} | {2430, ...} | A?????? | { } |
| 66 | 1000010 | {1,4,1,1} | {5670, ...} | A?????? | { } |
| 67 | 1000011 | {1,4,2} | {4050, ...} | A?????? | { } |
| 68 | 1000100 | {1,3,1,2} | {13230, ...} | A?????? | { } |
| 69 | 1000101 | {1,3,1,1,1} | {20790, ...} | A?????? | { } |
| 70 | 1000110 | {1,3,2,1} | {9450, ...} | A?????? | { } |
| 71 | 1000111 | {1,3,3} | {6750, ...} | A?????? | { } |
| 72 | 1001000 | {1,2,1,3} | {30870, ...} | A?????? | { } |
| 73 | 1001001 | {1,2,1,2,1} | {48510, ...} | A?????? | { } |
| 74 | 1001010 | {1,2,1,1,1,1} | {90090, ...} | A?????? | { } |
| 75 | 1001011 | {1,2,1,1,2} | {76230, ...} | A?????? | { } |
| 76 | 1001100 | {1,2,2,2} | {22050, ...} | A?????? | { } |
| 77 | 1001101 | {1,2,2,1,1} | {34650, ...} | A?????? | { } |
| 78 | 1001110 | {1,2,3,1} | {15750, ...} | A?????? | { } |
| 79 | 1001111 | {1,2,4} | {11250, ...} | A?????? | { } |
| 80 | 1010000 | {1,1,1,4} | {72030, ...} | A?????? | { } |
| 81 | 1010001 | {1,1,1,3,1} | {113190, ...} | A?????? | { } |
| 82 | 1010010 | {1,1,1,2,1,1} | {210210, ...} | A?????? | { } |
| 83 | 1010011 | {1,1,1,2,2} | {177870, ...} | A?????? | { } |
| 84 | 1010100 | {1,1,1,1,1,2} | {390390, ...} | A?????? | { } |
| 85 | 1010101 | {1,1,1,1,1,1,1} | {510510, 4849845, 37182145, 215656441, 955049953, 3212440751, 10131543907, ...} | A046325 | { } |
| 86 | 1010110 | {1,1,1,1,2,1} | {330330, ...} | A?????? | { } |
| 87 | 1010111 | {1,1,1,1,3} | {279510, ...} | A?????? | { } |
| 88 | 1011000 | {1,1,2,3} | {51450, ...} | A?????? | { } |
| 89 | 1011001 | {1,1,2,2,1} | {80850, ...} | A?????? | { } |
| 90 | 1011010 | {1,1,2,1,1,1} | {150150, ...} | A?????? | { } |
| 91 | 1011011 | {1,1,2,1,2} | {127050, ...} | A?????? | { } |
| 92 | 1011100 | {1,1,3,2} | {36750, ...} | A?????? | { } |
| 93 | 1011101 | {1,1,3,1,1} | {57750, ...} | A?????? | { } |
| 94 | 1011110 | {1,1,4,1} | {26250, ...} | A?????? | { } |
| 95 | 1011111 | {1,1,5} | {18750, ...} | A?????? | { } |
| 96 | 1100000 | {2,5} | {972, ...} | A?????? | { } |
| 97 | 1100001 | {2,4,1} | {1620, ...} | A?????? | { } |
| 98 | 1100010 | {2,3,1,1} | {3780, ...} | A?????? | { } |
| 99 | 1100011 | {2,3,2} | {2700, ...} | A?????? | { } |
| 100 | 1100100 | {2,2,1,2} | {8820, ...} | A?????? | { } |
| 101 | 1100101 | {2,2,1,1,1} | {13860, ...} | A?????? | { } |
| 102 | 1100110 | {2,2,2,1} | {6300, ...} | A?????? | { } |
| 103 | 1100111 | {2,2,3} | {4500, ...} | A?????? | { } |
| 104 | 1101000 | {2,1,1,3} | {20580, ...} | A?????? | { } |
| 105 | 1101001 | {2,1,1,2,1} | {32340, ...} | A?????? | { } |
| 106 | 1101010 | {2,1,1,1,1,1} | {60060, ...} | A?????? | { } |
| 107 | 1101011 | {2,1,1,1,2} | {50820, ...} | A?????? | { } |
| 108 | 1101100 | {2,1,2,2} | {14700, ...} | A?????? | { } |
| 109 | 1101101 | {2,1,2,1,1} | {23100, ...} | A?????? | { } |
| 110 | 1101110 | {2,1,3,1} | {10500, ...} | A?????? | { } |
| 111 | 1101111 | {2,1,4} | {7500, ...} | A?????? | { } |
| 112 | 1110000 | {3,4} | {648, ...} | A?????? | { } |
| 113 | 1110001 | {3,3,1} | {1080, ...} | A?????? | { } |
| 114 | 1110010 | {3,2,1,1} | {2520, ...} | A?????? | { } |
| 115 | 1110011 | {3,2,2} | {1800, ...} | A?????? | { } |
| 116 | 1110100 | {3,1,1,2} | {5880, ...} | A?????? | { } |
| 117 | 1110101 | {3,1,1,1,1} | {9240, ...} | A?????? | { } |
| 118 | 1110110 | {3,1,2,1} | {4200, ...} | A?????? | { } |
| 119 | 1110111 | {3,1,3} | {3000, ...} | A?????? | { } |
| 120 | 1111000 | {4,3} | {432, ...} | A?????? | { } |
| 121 | 1111001 | {4,2,1} | {720, ...} | A?????? | { } |
| 122 | 1111010 | {4,1,1,1} | {1680, ...} | A?????? | { } |
| 123 | 1111011 | {4,1,2} | {1200, ...} | A?????? | { } |
| 124 | 1111100 | {5,2} | {288, ...} | A?????? | { } |
| 125 | 1111101 | {5,1,1} | {480, ...} | A?????? | { } |
| 126 | 1111110 | {6,1} | {192, ...} | A?????? | { } |
| 127 | 1111111 | {7} | {128, 2187, 78125, 823543, 19487171, 62748517, 410338673, 893871739, 3404825447, ...}
{{2}, {3}, {5}, {7}, {11}, {13}, {17}, {19}, {23}, {29}, {31}, {37}, {41}, {43}, {47}, {53}, {59}, {61}, {67}, {71}, {73}, {79}, {83}, {89}, ...} |
A092759 | { } |
Increasing least integers of ordered prime signatures graded by prime signatures ordered by increasing least integers of prime signatures
Increasing least integers of ordered prime signatures graded by prime signature, primes signatures in the order of increasing smallest numbers of prime signatures. The number of ordered prime signatures corresponding to a given prime signature is given by a multinomial coefficient.
| Prime signature | Ordered prime signatures | Least integers of ordered prime signatures | |
|---|---|---|---|
| 1 | { } | { } | 1 |
| 2 | {1} | {1} | 2 |
| 3 | {2} | {2} | 4 |
| 4 | {1,1} | {1,1} | 6 |
| 5 | {3} | {3} | 8 |
| 6 | {2,1} | {2,1}, {1,2} | 12, 18 |
| 7 | {4} | {4} | 16 |
| 8 | {3,1} | {3,1}, {1,3} | 24, 54 |
| 9 | {1,1,1} | {1,1,1} | 30 |
| 10 | {5} | {5} | 32 |
| 11 | {2,2} | {2,2} | 36 |
| 12 | {4,1} | {4,1}, {1,4} | 48, 162 |
| 13 | {2,1,1} | {2,1,1}, {1,2,1}, {1,1,2} | 60, 90, 150 |
| 14 | {6} | {6} | 64 |
| 15 | {3,2} | {3,2}, {2,3} | 72, 108 |
| 16 | {5,1} | {5,1}, {1,5} | 96, 486 |
| 17 | {3,1,1} | {3,1,1}, {1,3,1}, {1,1,3} | 120, 270, 750 |
| 18 | {7} | {7} | 128 |
| 19 | {4,2} | {4,2}, {2,4} | 144, 324 |
| 20 | {2,2,1} | {2,2,1}, {2,1,2}, {1,2,2} | 180, 300, 450 |
| 21 | {6,1} | {6,1}, {1,6} | 192, 1458 |
| 22 | {1,1,1,1} | {1,1,1,1} | 210 |
| 23 | {3,3} | {3,3} | 216 |
| 24 | {4,1,1} | {4,1,1}, {1,4,1}, {1,1,4} | 240, 810, 3750 |
| 25 | {8} | {8} | 256 |
| 26 | {5,2} | {5,2}, {2,5} | 288, 972 |
| 27 | {3,2,1} | {3,2,1}, {2,3,1}, {3,1,2}, {2,1,3}, {1,3,2}, {1,2,3} | 360, 540, 600, 1350, 1500, 2250 |
| 28 | {7,1} | {7,1}, {1,7} | 384, 4374 |
| 29 | {2,1,1,1} | {2,1,1,1}, {1,2,1,1}, {1,1,2,1}, {1,1,1,2} | 420, 630, 1050, 1470 |
| 30 | {4,3} | {4,3}, {3,4} | 432, 648 |
| 31 | {5,1,1} | {5,1,1}, {1,5,1}, {1,1,5} | 480, 2430, 18750 |
| 32 | {9} | {9} | 512 |
| 33 | {6,2} | {6,2}, {2,6} | 576, 2916 |
| 34 | {4,2,1} | {4,2,1}, {4,1,2}, {2,4,1}, {1,4,2}, {2,1,4}, {1,2,4} | 720, 1200, 1620, 4050, 7500, 11250 |
| 35 | {8,1} | {8,1}, {1,8} | 768, 13122 |
| 36 | {3,1,1,1} | {3,1,1,1}, {1,3,1,1}, {1,1,3,1}, {1,1,1,3} | 840, 1890, 5250, 10290 |
| 37 | {5,3} | {5,3}, {3,5} | 864, 1944 |
| 38 | {2,2,2} | {2,2,2} | 900 |
| 39 | {6,1,1} | {6,1,1}, {1,6,1}, {1,1,6} | 960, 7290, 93750 |
| 40 | {10} | {10} | 1024 |
| 41 | {3,3,1} | {3,3,1}, {3,1,3}, {1,3,3} | 1080, 3000, 6750 |
| 42 | {7,2} | {7,2}, {2,7} | 1152, 8748 |
| 43 | {2,2,1,1} | {2,2,1,1}, {2,1,2,1}, {2,1,1,2}, {1,2,2,1}, {1,2,1,2}, {1,1,2,2} | 1260, 2100, 2940, 3150, 4410, 7350 |
| 44 | {4,4} | {4,4} | 1296 |
| 45 | {5,2,1} | {5,2,1}, {5,1,2}, {2,5,1}, {1,5,2}, {2,1,5}, {1,2,5} | 1440, 2400, 4860, 12150, 37500, 56250 |
| 46 | {9,1} | {9,1}, {1,9} | 1536, 39366 |
| 47 | {4,1,1,1} | {4,1,1,1}, {1,4,1,1}, {1,1,4,1}, {1,1,1,4} | 1680, 5670, 26250, 72030 |
| 48 | {6,3} | {6,3}, {3,6} | 1728, 5832 |
| 49 | {3,2,2} | {3,2,2}, {2,3,2}, {2,2,3} | 1800, 2700, 4500 |
| 50 | {7,1,1} | {7,1,1}, {1,7,1}, {1,1,7} | 1920, 21870, 468750 |
Ordered prime signatures in graded colexicographic order
...
Ordered prime signatures in the order of increasing smallest numbers of ordered prime signatures
1 1 2 2 3 4 4 6 5 8 6 12 7 16 8 18 9 24 10 30 11 32 12 36 13 48 14 54 15 60 16 64 17 72 18 90 19 96 20 108 21 120 22 128 23 144 24 150 25 162 26 180 27 192 28 210 29 216 30 240 31 256 32 270 33 288 34 300 35 324 36 360 37 384 38 420 39 432 40 450 41 480 42 486 43 512 44 540 45 576 46 600 47 630 48 648 49 720 50 750
Sequences
Least integer of each ordered prime signatures. (Cf. A055932)
- {1, 2, 4, 6, 8, 12, 16, 18, 24, 30, 32, 36, 48, 54, 60, 64, 72, 90, 96, 108, 120, 128, 144, 150, 162, 180, 192, 210, 216, 240, 256, 270, 288, 300, 324, 360, 384, 420, 432, 450, 480, 486, 512, 540, 576, ...}
Least integer of each ordered prime signatures (A055932) arranged by prime signature (each row starting with least integer of each prime signature, A025487). (Cf. A096903)
- {1, 2, 4, 6, 8, 12, 18, 16, 24, 54, 30, 32, 36, 48, 162, 60, 90, 150, 64, 72, 108, 96, 486, 120, 270, 750, 128, 144, 324, 180, 300, 450, 192, 1458, 210, 216, 240, 810, 3750, 256, 288, 972, 360, 540, 600, ...}
Smallest number with same sequence of exponents in canonical prime factorization as n. (Cf. A071364)
- {1, 2, 2, 4, 2, 6, 2, 8, 4, 6, 2, 12, 2, 6, 6, 16, 2, 18, 2, 12, 6, 6, 2, 24, 4, 6, 8, 12, 2, 30, 2, 32, 6, 6, 6, 36, 2, 6, 6, 24, 2, 30, 2, 12, 12, 6, 2, 48, 4, 18, 6, 12, 2, 54, 6, 24, 6, 6, 2, 60, 2, 6, 12, 64, 6, 30, ...}
See also
- Ordered prime signatures
- Orderings of ordered prime signatures
- Compositions
- Orderings of compositions
Notes
- ↑ We should consider having the empty sum here, the leading zeros not being normally represented (we put the leading zero for zero only to avoid an empty representation for zero, which would not be convenient.
