# interesting prime-related integer sequences

Here are some interesting prime-related integer sequences that can be easily represented as a set. I have copied them from The On-Line Encyclopedia of Integer Sequences® (OEIS®) but I have written the formulas of the sets myself.

`p`, `q`, `r`, `s`, `t` and `u` denote distinct primes.

### table of contents

- one prime factor (Ω(
`n`) = 1) - two prime factors (Ω(
`n`) = 2) - three prime factors (Ω(
`n`) = 3) - four prime factors (Ω(
`n`) = 4) - five prime factors (Ω(
`n`) = 5) - six prime factors (Ω(
`n`) = 6)

## one prime factor (Ω(`n`) = 1)

OEIS | set | original description | smallest items |
---|---|---|---|

`A000040` |
{p} |
The prime numbers. | 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 |

## two prime factors (Ω(`n`) = 2)

OEIS | set | original description | smallest items |
---|---|---|---|

`A001248` |
{p^{2}} |
Squares of primes. | 4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481 |

`A006881` |
{pq} |
Squarefree semiprimes: Numbers that are the product of two distinct primes. | 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85 |

`A001358` |
{p^{2}, pq} |
Semiprimes (or biprimes): products of two primes. | 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69 |

## three prime factors (Ω(`n`) = 3)

OEIS | set | original description | smallest items |
---|---|---|---|

`A030078` |
{p^{3}} |
Cubes of primes. | 8, 27, 125, 343, 1331, 2197, 4913, 6859, 12167, 24389, 29791, 50653, 68921, 79507 |

`A054753` |
{p^{2}q} |
Numbers which are the product of a prime and the square of a different prime. | 12, 18, 20, 28, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99, 116, 117, 124, 147, 148, 153 |

`A007304` |
{pqr} |
Sphenic numbers: products of 3 distinct primes. | 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195 |

`A285508` |
{p^{3}, p^{2}q} |
Numbers with exactly three prime factors, not all distinct. | 8, 12, 18, 20, 27, 28, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99, 116, 117, 124, 125, 147 |

`A217856` |
{p^{2}q, pqr} |
Numbers with three prime factors, not necessarily distinct, except cubes of primes. | 12, 18, 20, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, 102, 105 |

`A014612` |
{p^{3}, p^{2}q, pqr} |
Numbers that are the product of exactly three (not necessarily distinct) primes. | 8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, 102 |

## four prime factors (Ω(`n`) = 4)

OEIS | set | original description | smallest items |
---|---|---|---|

`A030514` |
{p^{4}} |
4th powers of primes. | 16, 81, 625, 2401, 14641, 28561, 83521, 130321, 279841, 707281, 923521, 1874161 |

`A065036` |
{p^{3}q} |
Product of the cube of a prime (A030078) and a different prime. | 24, 40, 54, 56, 88, 104, 135, 136, 152, 184, 189, 232, 248, 250, 296, 297, 328, 344, 351 |

`A085986` |
{p^{2}q^{2}} |
Squares of the squarefree semiprimes. | 36, 100, 196, 225, 441, 484, 676, 1089, 1156, 1225, 1444, 1521, 2116, 2601, 3025 |

`A085987` |
{p^{2}qr} |
Product of exactly four primes, three of which are distinct. | 60, 84, 90, 126, 132, 140, 150, 156, 198, 204, 220, 228, 234, 260, 276, 294, 306, 308 |

`A046386` |
{pqrs} |
Products of four distinct primes. | 210, 330, 390, 462, 510, 546, 570, 690, 714, 770, 798, 858, 870, 910, 930, 966, 1110 |

`A014613` |
{p^{4}, p^{3}q, p^{2}q^{2}, p^{2}qr, pqrs} |
Numbers that are products of 4 primes (these numbers are sometimes called "4-almost primes", a generalization of semiprimes). | 16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, 104, 126, 132, 135, 136, 140, 150, 152 |

## five prime factors (Ω(`n`) = 5)

OEIS | set | original description | smallest items |
---|---|---|---|

`A050997` |
{p^{5}} |
Fifth powers of primes. | 32, 243, 3125, 16807, 161051, 371293, 1419857, 2476099, 6436343, 20511149 |

`A178739` |
{p^{4}q} |
Product of the 4th power of a prime (A030514) and a different prime. | 48, 80, 112, 162, 176, 208, 272, 304, 368, 405, 464, 496, 567, 592, 656, 688, 752, 848 |

`A143610` |
{p^{3}q^{2}} |
Numbers of the form p^2*q^3, where p,q are distinct primes. | 72, 108, 200, 392, 500, 675, 968, 1125, 1323, 1352, 1372, 2312, 2888, 3087, 3267 |

`A189975` |
{p^{3}qr} |
Numbers with prime factorization pqr^3. | 120, 168, 264, 270, 280, 312, 378, 408, 440, 456, 520, 552, 594, 616, 680, 696, 702 |

`A179643` |
{p^{2}q^{2}r} |
Products of exactly 2 distinct primes squares and a different prime (p^2*q^2*r). | 180, 252, 300, 396, 450, 468, 588, 612, 684, 700, 828, 882, 980, 1044, 1100, 1116 |

`A046387` |
{pqrst} |
Products of 5 distinct primes. | 2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6270, 6510, 6630, 7410, 7590 |

`A014614` |
{p^{5}, p^{4}q, p^{3}q^{2}, p^{3}qr, p^{2}q^{2}r, p^{2}qrs, pqrst} |
Numbers that are products of 5 primes (or 5-almost primes, a generalization of semiprimes). | 32, 48, 72, 80, 108, 112, 120, 162, 168, 176, 180, 200, 208, 243, 252, 264, 270, 272 |

## six prime factors (Ω(`n`) = 6)

OEIS | set | original description | smallest items |
---|---|---|---|

`A030516` |
{p^{6}} |
Numbers with 7 divisors. | 64, 729, 15625, 117649, 1771561, 4826809, 24137569, 47045881, 148035889 |

`A178740` |
{p^{5}q} |
Product of the 5th power of a prime (A050997) and a different prime. | 96, 160, 224, 352, 416, 486, 544, 608, 736, 928, 992, 1184, 1215, 1312, 1376, 1504 |

`A189988` |
{p^{4}q^{2}} |
Numbers with prime factorization p^2*q^4. | 144, 324, 400, 784, 1936, 2025, 2500, 2704, 3969, 4624, 5625, 5776, 8464, 9604, 9801 |

`A179644` |
{p^{4}qr} |
Product of the 4th power of a prime and 2 different distinct primes (p^4*q*r). | 240, 336, 528, 560, 624, 810, 816, 880, 912, 1040, 1104, 1134, 1232, 1360, 1392, 1456 |

`A162142` |
{p^{3}q^{3}} |
Numbers that are the cube of a product of two distinct primes. | 216, 1000, 2744, 3375, 9261, 10648, 17576, 35937, 39304, 42875, 54872, 59319 |

`A163569` |
{p^{3}q^{2}r} |
Numbers of the form p^3*q^2*r where p, q and r are three distinct primes. | 360, 504, 540, 600, 756, 792, 936, 1176, 1188, 1224, 1350, 1368, 1400, 1404, 1500 |

`A162143` |
{p^{2}q^{2}r^{2}} |
a(n) = A007304(n)^2. | 900, 1764, 4356, 4900, 6084, 10404, 11025, 12100, 12996, 16900, 19044, 23716 |

`A067885` |
{pqrstu} |
Product of 6 distinct primes. | 30030, 39270, 43890, 46410, 51870, 53130, 62790, 66990, 67830, 71610, 72930 |

`A046306` |
{p^{6}, p^{5}q, p^{4}q^{2}, p^{4}qr, p^{3}q^{3}, p^{3}q^{2}r, p^{3}qrs, p^{2}q^{2}r^{2}, p^{2}q^{2}rs, p^{2}qrst, pqrstu} |
Numbers that are divisible by exactly 6 primes with multiplicity. | 64, 96, 144, 160, 216, 224, 240, 324, 336, 352, 360, 400, 416, 486, 504, 528, 540, 544 |