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Template:Table of prime signatures in the order of increasing smallest numbers of prime signatures

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Prime signatures in the order of increasing smallest numbers of prime signatures and their numbers
n
Prime

signature

Sequence A-number Description
1 { } {1} Unit
2 {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, ...} A000040 Prime numbers
3 {2} {4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, ...} A001248 Squares of prime numbers (squarefull biprimes)
4 {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, ...} A006881 Two distinct prime factors (squarefree biprimes)
5 {3} {8, 27, 125, 343, 1331, 2197, 4913, 6859, 12167, 24389, 29791, 50653, 68921, 79507, ...} A030078 Cubes of prime numbers (cubefull triprimes)
6 {2,1} {12, 18, 20, 28, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99, 116, 117, 124, 147, 148, 153, ...} A054753  
7 {4} {16, 81, 625, 2401, 14641, 28561, 83521, 130321, 279841, 707281, 923521, 1874161, ...} A030514 Fourth powers of prime numbers (biquadratefull tetraprimes)
8 {3,1} {24, 40, 54, 56, 88, 104, 135, 136, 152, 184, 189, 232, 248, 250, 296, 297, 328, 344, 351, ...} A065036  
9 {1,1,1} {30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, ...} A007304 Three distinct prime factors (sphenic numbers) (squarefree triprimes)
10 {5} {32, 243, 3125, 16807, 161051, 371293, 1419857, 2476099, 6436343, 20511149, 28629151, ...} A050997 Fifth powers of primes (powerfull pentaprimes)
11 {2,2} {36, 100, 196, 225, 441, 484, 676, 1089, 1156, 1225, 1444, 1521, 2116, 2601, 3025, 3249, ...} A085986 Squares of squarefree biprimes
12 {4,1} {48, 80, 112, 162, 176, 208, 272, 304, 368, 405, 464, 496, 567, 592, 656, 688, 752, 848, ...} A178739  
13 {2,1,1} {60, 84, 90, 126, 132, 140, 150, 156, 198, 204, 220, 228, 234, 260, 276, 294, 306, 308, 315, ...} A085987  
14 {6} {64, 729, 15625, 117649, 1771561, 4826809, 24137569, 47045881, 148035889, 594823321, ...} A030516 Sixth powers of primes (powerfull hexaprimes)
15 {3,2} {72, 108, 200, 392, 500, 675, 968, 1125, 1323, 1352, 1372, 2312, 2888, 3087, 3267, 4232, ...} A143610 (subset of Achilles numbers)
16 {5,1} {96, 160, 224, 352, 416, 486, 544, 608, 736, 928, 992, 1184, 1215, 1312, 1376, 1504, 1696, ...} A178740  
17 {3,1,1} {120, 168, 180, 252, 264, 270, 280, 300, 312, 378, 396, 408, 440, 450, 456, 468, 520, ...} A179642  
18 {7} {128, 2187, 78125, 823543, 19487171, 62748517, 410338673, 893871739, 3404825447, ...} A092759 Seventh powers of primes (powerfull heptaprimes)
19 {4,2} {144, 324, 400, 784, 1936, 2025, 2500, 2704, 3969, 4624, 5625, 5776, 8464, 9604, 9801, ...} NOT QUITE A030633

15 divisors

{4,2} and {14}

(subset of square numbers)
20 {2,2,1} {180, 252, 300, 396, 450, 468, 588, 612, 684, 700, 828, 882, 980, 1044, 1100, 1116, 1300, ...} A179643  
21 {6,1} {192, 320, 448, 704, 832, 1088, 1216, 1458, 1472, 1856, 1984, 2368, 2624, 2752, 3008, 3392, ...} NOT QUITE A030632

14 divisors

{6,1} and {13}

 
22 {1,1,1,1} {210, 330, 390, 462, 510, 546, 570, 690, 714, 770, 798, 858, 870, 910, 930, 966, 1110, 1122, ...} A046386 Four distinct prime factors (squarefree tetraprimes)
23 {3,3} {216, 1000, 2744, 3375, 9261, 10648, 17576, 35937, 39304, 42875, 54872, 59319, 97336, ...} A162142 Cubes of squarefree biprimes
24 {4,1,1} {240, 336, 528, 560, 624, 810, 816, 880, 912, 1040, 1104, 1134, 1232, 1360, 1392, 1456, ...} A179644  
25 {8} {256, 6561, 390625, 5764801, 214358881, 815730721, 6975757441, 16983563041, ...} A179645 Eighth powers of primes (powerfull octaprimes)
26 {5,2} {288, 800, 972, 1568, 3872, 5408, 6075, 9248, 11552, 11907, 12500, 16928, 26912, 28125, ...} A179646 (subset of Achilles numbers)
27 {3,2,1} {360, 504, 540, 600, 756, 792, 936, 1176, 1188, 1224, 1350, 1368, 1400, 1404, 1500, 1656, ...} A163569  
28 {7,1} {384, 640, 896, 1408, 1664, 2176, 2432, 2944, 3712, 3968, 4374, 4736, 5248, 5504, 6016, ...} A179664  
29 {2,1,1,1} {420, 630, 1050, 1470, ...} A??????  
30 {4,3} {432, 648, 2000, 5000, 5488, 10125, 16875, 19208, 21296, 27783, 35152, 64827, 78608, ...} A179666 (subset of Achilles numbers)
31 {5,1,1} {480, 672, 1056, 1120, 1248, 1632, 1760, 1824, 2080, 2208, 2430, 2464, 2720, 2784, 2912, ...} A179667  
32 {9} {512, 19683, 1953125, 40353607, 2357947691, 10604499373, 118587876497, 322687697779, ...} A179665 Ninth powers of primes (powerfull nonaprimes)
33 {6,2} {576, 1600, 2916, 3136, 7744, 10816, 18225, 18496, 23104, 33856, 35721, 53824, 61504, ...} NOT QUITE A137484

21 divisors

{6,2} and {20}

(subset of square numbers)
34 {4,2,1} {720, 1008, 1200, 1584, 1620, 1872, 2268, 2352, 2448, 2736, 2800, 3312, 3564, 3920, 4050, ...} A179669  
35 {8,1} {768, 13122, 781250, 1171875, ...} A??????  
36 {3,1,1,1} {840, 1320, 1560, 1848, 1890, 2040, 2184, 2280, 2760, 2856, 2970, 3080, 3192, 3432, 3480, ...} A179670  
37 {5,3} {864, 1944, 4000, 10976, 25000, 30375, 42592, 70304, 83349, 84375, 134456, 157216, ...} A179671 (subset of Achilles numbers)
38 {2,2,2} {900, 1764, 4356, 4900, 6084, 10404, 11025, 12100, 12996, 16900, 19044, 23716, 27225, ...} A162143 Squares of squarefree triprimes
39 {6,1,1} {960, 1344, 2112, 2240, 2496, 3264, 3520, 3648, 4160, 4416, 4928, 5440, 5568, 5824, 5952, ...} A179672  
40 {10} {1024, 59049, 9765625, 282475249, 25937424601, 137858491849, 2015993900449, ...} A030629 Tenth powers of primes (powerfull decaprimes)
41 {3,3,1} {1080, 1512, 2376, 2808, 3000, 3672, 4104, 4968, 6264, 6696, 6750, 7000, 7992, 8232, ...} A179688  
42 {7,2} {1152, 8748, 312500, 703125, ...} A?????? (subset of Achilles numbers)
43 {2,2,1,1} {1260, 1980, 2100, 2340, 2772, 2940, 3060, 3150, 3276, 3300, 3420, 3900, 4140, 4284, ...} A179690  
44 {4,4} {1296, 10000, 38416, 50625, 194481, 234256, 456976, 1185921, 1336336, 1500625, ...} NOT QUITE A137488

25 divisors

{4,4} and {24}

Biquadrates of squarefree biprimes
45 {5,2,1} {1440, 2016, 2400, 3168, 3744, 4704, 4860, 4896, 5472, 5600, 6624, 6804, 7840, 8352, ...} A179691  
46 {9,1} {1536, 39366, 3906250, 5859375, ...} A??????  
47 {4,1,1,1} {1680, 2640, 3120, 3696, 4080, 4368, 4560, 5520, 5670, 5712, 6160, 6384, 6864, 6960, ...} A179693  
48 {6,3} {1728, ...} A?????? (subset of cube numbers)
49 {3,2,2} {1800, 2700, 3528, 4500, 5292, 8712, 9800, 12168, 12348, 13068, 18252, 20808, 24200, ...} A179695 (subset of Achilles numbers)
50 {7,1,1} {1920, ...} A??????  
51 {11} {2048, 177147, 48828125, 1977326743, 285311670611, 1792160394037, 34271896307633, ...} A079395 Eleventh powers of primes (powerfull hendecaprimes)
52 {4,3,1} {2160, 3024, 3240, 4536, 4752, 5616, 6000, 7128, 7344, 8208, 8424, 9936, 11016, 12312, ...} A179698  
53 {8,2} {2304, 26244, ...} A?????? (subset of square numbers)
54 {1,1,1,1,1} {2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6270, 6510, 6630, 7410, 7590, 7770, ...} A046387 Five distinct prime factors (squarefree pentaprimes)
55 {3,2,1,1} {2520, 3780, 3960, 4200, 4680, 5544, 5880, 5940, 6120, 6552, 6600, 6840, 7020, 7800, 8280, ...} A179700  
56 {5,4} {2592, ...} A?????? (subset of Achilles numbers)
57 {6,2,1} {2880, ...} A??????  
58 {10,1} {3072, 118098, 19531250, 29296875, ...} A??????  
59 {5,1,1,1} {3360, 17010, ...} A??????  
60 {7,3} {3456, ...} A?????? (subset of Achilles numbers)
61 {4,2,2} {3600, 8100, ...} A?????? (subset of square numbers)
62 {8,1,1} {3840, ...} A??????  
63 {12} {4096, 531441, 244140625, 13841287201, 3138428376721, 23298085122481, ...} A030631 Twelfth powers of primes (powerfull dodecaprimes)
64 {5,3,1} {4320, ...} A??????  
65 {9,2} {4608, 78732, ...} A?????? (subset of Achilles numbers)
66 {2,1,1,1,1} {4620, 6930, ...} A??????  
67 {4,2,1,1} {5040, 11340, ...} A??????  
68 {6,4} {5184, ...} A?????? (subset of square numbers)
69 {3,3,2} {5400, 13500, ...} A?????? (subset of Achilles numbers)
70 {7,2,1} {5760, ...} A??????  
71 {11,1} {6144, 354294, ...} A??????  
72 {2,2,2,1} {6300, 22050, ...} A??????  
73 {4,4,1} {6480, ...} A??????  
74 {6,1,1,1} {6720, ...} A??????  
75 {8,3} {6912, ...} A?????? (subset of Achilles numbers)
76 {5,2,2} {7200, ...} A?????? (subset of Achilles numbers)
77 {3,3,1,1} {7560, ...} A??????  
78 {9,1,1} {7680, ...} A??????  
79 {5,5} {7776, 100000, 759375, ...} A?????? Fifth powers of squarefree biprimes
80 {13} {8192, 1594323, 1220703125, 96889010407, 34522712143931, 302875106592253, ...} A138031 Thirteenth powers of primes (powerfull tridecaprimes)
81 {6,3,1} {8640, ...} A??????  
82 {10,2} {9216, ...} A?????? (subset of square numbers)
83 {3,1,1,1,1} {9240, ...} A??????  
84 {5,2,1,1} {10080, ...} A??????  
85 {7,4} {10368, ...} A?????? (subset of Achilles numbers)
86 {4,3,2} {10800, ...} A?????? (subset of Achilles numbers)
87 {8,2,1} {11520, ...} A??????  
88 {12,1} {12288, 1062882, ...} A??????  
89 {3,2,2,1} {12600, ...} A??????  
90 {5,4,1} {12960, ...} A??????  
91 {7,1,1,1} {13440, ...} A??????  
92 {9,3} {13824, ...} A?????? (subset of cube numbers)
93 {2,2,1,1,1} {13860, ...} A??????  
94 {6,2,2} {14400, ...} A?????? (subset of square numbers)
95 {4,3,1,1} {15120, ...} A??????  
96 {10,1,1} {15360, ...} A??????  
97 {6,5} {15552, ...} A?????? (subset of Achilles numbers)
98 {14} {16384, 4782969, 6103515625, ...} A?????? Fourteenth powers of primes (powerfull tetradecaprimes)