Orderings of prime signatures

Prime signatures in graded colexicographic order

In this table, the prime signatures appear in graded colexicographic order, i.e. graded by increasing 

Ω (n)

, i.e. number of prime factors (with repetition), then by colexicographic order of exponents (which nicely sorts by increasing 

ω (n)

, i.e. number of distinct prime factors). With this convention, the prime signature internal order corresponds to the order of exponents of the smallest number with a given prime signature, where the prime factors are commonly listed from smallest to largest.

Prime signatures in graded colexicographic order 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	{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)
8	{4}	{16, 81, 625, 2401, 14641, 28561, 83521, 130321, 279841, 707281, 923521, 1874161, ...}	A030514	Fourth powers of prime numbers (biquadratefull tetraprimes)
9	{3,1}	{24, 40, 54, 56, 88, 104, 135, 136, 152, 184, 189, 232, 248, 250, 296, 297, 328, 344, 351, ...}	A065036
10	{2,2}	{36, 100, 196, 225, 441, 484, 676, 1089, 1156, 1225, 1444, 1521, 2116, 2601, 3025, 3249, ...}	A085986	Squares of squarefree biprimes
11	{2,1,1}	{60, 84, 90, 126, 132, 140, 150, 156, 198, 204, 220, 228, 234, 260, 276, 294, 306, 308, 315, ...}	A085987
12	{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)
13	{5}	{32, 243, 3125, 16807, 161051, 371293, 1419857, 2476099, 6436343, 20511149, 28629151, ...}	A050997	Fifth powers of primes (powerfull pentaprimes)
14	{4,1}	{48, 80, 112, 162, 176, 208, 272, 304, 368, 405, 464, 496, 567, 592, 656, 688, 752, 848, ...}	A178739
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	{3,1,1}	{120, 168, 180, 252, 264, 270, 280, 300, 312, 378, 396, 408, 440, 450, 456, 468, 520, ...}	A179642
17	{2,2,1}	{180, 252, 300, 396, 450, 468, 588, 612, 684, 700, 828, 882, 980, 1044, 1100, 1116, 1300, ...}	A179643
18	{2,1,1,1}	{420, 630, 1050, 1470, ...}	A??????
19	{1,1,1,1,1}	{2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6270, 6510, 6630, 7410, 7590, ...}	A046387	Five distinct prime factors (squarefree pentaprimes)
20	{6}	{64, 729, 15625, 117649, 1771561, 4826809, 24137569, 47045881, 148035889, 594823321, ...}	A030516	Sixth powers of primes (powerfull hexaprimes)
21	{5,1}	{96, 160, 224, 352, 416, 486, 544, 608, 736, 928, 992, 1184, 1215, 1312, 1376, 1504, 1696, ...}	A178740
22	{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)
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	{3,2,1}	{360, 504, 540, 600, 756, 792, 936, 1176, 1188, 1224, 1350, 1368, 1400, 1404, 1500, 1656, ...}	A163569
26	{2,2,2}	{900, 1764, 4356, 4900, 6084, 10404, 11025, 12100, 12996, 16900, 19044, 23716, 27225, ...}	A162143	Squares of squarefree triprimes
27	{3,1,1,1}	{840, 1320, 1560, 1848, 1890, 2040, 2184, 2280, 2760, 2856, 2970, 3080, 3192, 3432, 3480, ...}	A179670
28	{2,2,1,1}	{1260, 1980, 2100, 2340, 2772, 2940, 3060, 3150, 3276, 3300, 3420, 3900, 4140, 4284, ...}	A179690
29	{2,1,1,1,1}	{4620, 6930, ...}	A??????
30	{1,1,1,1,1,1}	{30030, 39270, 43890, 46410, 51870, 53130, 62790, 66990, 67830, 71610, 72930, 79170, ...}	A067885	Six distinct prime factors (squarefree hexaprimes)
31	{7}	{128, 2187, 78125, 823543, 19487171, 62748517, 410338673, 893871739, 3404825447, ...}	A092759	Seventh powers of primes (powerfull heptaprimes)
32	{6,1}	{192, 320, 448, 704, 832, 1088, 1216, 1458, 1472, 1856, 1984, 2368, 2624, 2752, 3008, ...}	NOT QUITE A030632 14 divisors {6,1} and {13}
33	{5,2}	{288, 800, 972, 1568, 3872, 5408, 6075, 9248, 11552, 11907, 12500, 16928, 26912, 28125, ...}	A179646	(subset of Achilles numbers)
34	{4,3}	{432, 648, 2000, 5000, 5488, 10125, 16875, 19208, 21296, 27783, 35152, 64827, 78608, ...}	A179666	(subset of Achilles numbers)
35	{5,1,1}	{480, 672, 1056, 1120, 1248, 1632, 1760, 1824, 2080, 2208, 2430, 2464, 2720, 2784, 2912, ...}	A179667
36	{4,2,1}	{720, 1008, 1200, 1584, 1620, 1872, 2268, 2352, 2448, 2736, 2800, 3312, 3564, 3920, 4050, ...}	A179669
37	{3,3,1}	{1080, 1512, 2376, 2808, 3000, 3672, 4104, 4968, 6264, 6696, 6750, 7000, 7992, 8232, ...}	A179688
38	{3,2,2}	{1800, 2700, 3528, 4500, 5292, 8712, 9800, 12168, 12348, 13068, 18252, 20808, 24200, ...}	A179695	(subset of Achilles numbers)
39	{4,1,1,1}	{1680, 2640, 3120, 3696, 4080, 4368, 4560, 5520, 5670, 5712, 6160, 6384, 6864, 6960, ...}	A179693
40	{3,2,1,1}	{2520, 3780, 3960, 4200, 4680, 5544, 5880, 5940, 6120, 6552, 6600, 6840, 7020, 7800, ...}	A179700
41	{2,2,2,1}	{6300, 22050, ...}	A??????
42	{3,1,1,1,1}	{9240, ...}	A??????
43	{2,2,1,1,1}	{13860, ...}	A??????
44	{2,1,1,1,1,1}	{60060, 450450, ...}	A??????
45	{1,1,1,1,1,1,1}	{510510, 4849845, 37182145, 215656441, 955049953, 3212440751, 10131543907, ...}	A046325	Seven distinct prime factors (squarefree heptaprimes)
46	{8}	{256, 6561, 390625, 5764801, 214358881, 815730721, 6975757441, 16983563041, ...}	A179645	Eighth powers of primes (powerfull octaprimes)
47	{7,1}	{384, 640, 896, 1408, 1664, 2176, 2432, 2944, 3712, 3968, 4374, 4736, 5248, 5504, 6016, ...}	A179664
48	{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)
49	{5,3}	{864, 1944, 4000, 10976, 25000, 30375, 42592, 70304, 83349, 84375, 134456, 157216, ...}	A179671	(subset of Achilles numbers)
50	{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
51	{6,1,1}	{960, 1344, 2112, 2240, 2496, 3264, 3520, 3648, 4160, 4416, 4928, 5440, 5568, 5824, 5952, ...}	A179672
52	{5,2,1}	{1440, 2016, 2400, 3168, 3744, 4704, 4860, 4896, 5472, 5600, 6624, 6804, 7840, 8352, ...}	A179691
53	{4,3,1}	{2160, 3024, 3240, 4536, 4752, 5616, 6000, 7128, 7344, 8208, 8424, 9936, 11016, 12312, ...}	A179698
54	{4,2,2}	{3600, 8100, ...}	A??????	(subset of square numbers)
55	{3,3,2}	{5400, 13500, ...}	A??????	(subset of Achilles numbers)
56	{5,1,1,1}	{3360, 17010, ...}	A??????
57	{4,2,1,1}	{5040, 11340, ...}	A??????
58	{3,3,1,1}	{7560, ...}	A??????
59	{3,2,2,1}	{12600, ...}	A??????
60	{2,2,2,2}	{44100, ...}	A??????	Squares of squarefree tetraprimes
61	{4,1,1,1,1}	{18480, ...}	A??????
62	{3,2,1,1,1}	{27720, ...}	A??????
63	{2,2,2,1,1}	{69300, ...}	A??????
64	{3,1,1,1,1,1}	{120120, ...}	A??????
65	{2,2,1,1,1,1}	{180180, ...}	A??????
66	{2,1,1,1,1,1,1}	{1021020, ...}	A??????
67	{1,1,1,1,1,1,1,1}	{9699690, 11741730, 13123110, 14804790, 15825810, 16546530, 17160990, 17687670, ...}	A123322	Eight distinct prime factors (squarefree octaprimes)

Prime signatures in the order of increasing smallest numbers of prime signatures

In this table, the prime signatures appear in the order of increasing smallest numbers of prime signatures. Note that the prime signatures are not graded by increasing 

Ω (n)

. The exponents are listed in decreasing order. With this convention, the prime signature internal order corresponds to the order of exponents of the smallest numbers of prime signatures, where the prime factors are commonly listed from smallest to largest.

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)

Sequences

A118914 Concatenation of the prime signatures (in increasing order of exponents of prime power components) of the positive integers.

 

{1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 4, 2, 1, 2, ...}

A036035 Least integer of each prime signature, in graded (colexicographic order or reflected colexicographic order) of exponents.

 

{1, 2, 4, 6, 8, 12, 30, 16, 24, 36, 60, 210, 32, 48, 72, 120, 180, 420, 2310, 64, 96, 144, 216, 240, 360, 900, 840, 1260, 4620, 30030, 128, 192, 288, 432, 480, 720, 1080, 1800, 1680, 2520, 6300, 9240, ...}

A046523 Smallest number with same prime signature as 

n

.

 

{1, 2, 2, 4, 2, 6, 2, 8, 4, 6, 2, 12, 2, 6, 6, 16, 2, 12, 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, 12, 6, 12, 2, 24, 6, 24, 6, 6, 2, 60, 2, 6, 12, 64, 6, 30, 2, ...}

A025487 Least integer of each prime signature; also products of primorial numbers A002110.

 

{1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 216, 240, 256, 288, 360, 384, 420, 432, 480, 512, 576, 720, 768, 840, 864, 900, 960, 1024, 1080, 1152, 1260, 1296, ...}

A181087 Prime signatures in the order of increasing smallest numbers of prime signatures.

 

{1, 2, 1, 1, 3, 1, 2, 4, 1, 3, 1, 1, 1, 5, 2, 2, 1, 4, 1, 1, 2, 6, 2, 3, 1, 5, 1, 1, 3, 7, 2, 4, 1, 2, 2, 1, 6, 1, 1, 1, 1, 3, 3, 1, 1, 4, 8, 2, 5, 1, 2, 3, 1, 7, 1, 1, 1, 2, 3, 4, 1, 1, 5, 9, 2, 6, 1, 2, 4, 1, 8, 1, 1, 1, 3, 3, 5, ...}

A095904 Triangular array of natural numbers (greater than 1) arranged by prime signature.

 

{2, 3, 4, 5, 9, 6, 7, 25, 10, 8, 11, 49, 14, 27, 12, 13, 121, 15, 125, 18, 16, 17, 169, 21, 343, 20, 81, 24, 19, 289, 22, 1331, 28, 625, 40, 30, 23, 361, 26, 2197, 44, 2401, 54, 42, 32, 29, 529, 33, 4913, 45, ...}

Sequences related to ordered prime signatures

The ordered prime signatures (where the order of exponents matters) correspond to compositions of 

Ω (n)

instead of partitions of 

Ω (n)

.

A055932 Least integer of each ordered prime signatures.

 

{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, ...}

A096903 Least integer of each ordered prime signatures (A055932) arranged by prime signature (each row starting with least integer of each prime signature, A025487).

 

{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, ...}

See also

• Prime signature
• Orderings

• A179665 

  p 9

  with 

  p

  prime.
• A179668 Products of the 

  8

  th power of a prime and a distinct prime 

  (p 8 q)

  .
• A179689 Products of 

  2

  distinct primes 

  (p 7q 2)

  .
• A179696 Products of 

  3

  distinct primes 

  (p 7q r)

  .
• A179694 Products of 

  2

  distinct primes 

  (p 6 q 3)

  .

• A030629 Numbers with 

  11

  divisors. (Prime(n)^10.)
• A179692 Products of 

  2

  distinct primes 

  (p 9 q)

  .
• A179699 Products of 

  2

  distinct primes 

  (p 8 q 2)

  .
• A179696 Numbers with prime signature 

  {7, 1, 1}

  , i.e. of form 

  p 7q r

  with 

  p, q

  and 

  r

  prime.

• A079395 Prime(n)^11.

• Ordered prime signature (where the order of exponents matters, corresponding to compositions of 

  Ω (n)

  )

External links

• Will Nicholes, List of the first 400 prime signatures, Copyright ©2010.
• Will Nicholes, Iterative mapping of prime signatures, Copyright ©2010.