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# Orderings of ordered prime signatures

## Binary representation ordering of the ordered prime signatures

${\displaystyle n\,}$ ${\displaystyle n_{2}\,}$ 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.

${\displaystyle n\,}$ 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

...

```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, ...}