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有序素数签名的排序

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有序素数签名的二进制表示排序

 
有序素数签名 数字

{不同的素因子}

OEIS公司

数

主要签名
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美元 { }
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号 { }
三 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号 { }
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号 { }
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型 { }
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号 { }
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型 { }
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号 { }
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号 { }
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型 { }
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美元 { }
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型 { }
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号 { }
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号 { }
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型 { }
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号 { }
86 1010110 {1,1,1,1,2,1} {330330, ...} A？？？？？？ { }
87 1010111 ｛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} {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 ｛2，2，1｝ {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号 { }


增加素数签名的最小整数

增加的最小整数有序素数签名分级依据主要签名，素数签名按素数签名的最小数目递增的顺序排列。与给定素数签名相对应的有序素数签名的数量由多项式系数.

 
主要签名 有序素数签名 有序素数签名的最小整数
1 { } { } 1
2 {1} {1} 2
三 {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｝，｛1，5｝ 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}，{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


分级排列序中的有序素数签名

...

有序素数签名以最小数量的有序素数签名递增的顺序排列

1 12 2个3 44 65 86 127 168 189 2410 3011 3212 3613 4814 5415 6016 6417 7218 9019 9620 10821 12022 12823 14424 15025 16226 18027 19228 21029 21630 24031 25632 27033 28834 30035 32436 36037 38438 42039 43240 45041 48042 48643 51244 54045 57646 60047 63048 64849 72050 750

序列

每个的最小整数有序素数签名（参见。A055932号)

{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, ...}

每个的最小整数有序素数签名(A055932号)由安排主要签名（每行以每行中的最小整数开始主要签名,A025487号). （参见。A096903号)

{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, ...}

正则素因式分解中指数序列与n相同的最小数。A071364号)

{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, ...}

另请参见



笔记

  1. ↑ 我们应该考虑空总和在这里前导零不正常表示（我们将前导零表示为零，只是为了避免零的表示为空，这不方便。
检索自“https://oeis.org/w/index.php？title=Orderings_of_ordered_prime_signatures&oldid=1557211"