# interesting prime-related integer sequences

Here are some integer sequences that can be easily expressed as a set.
`p`, `q` and `r` denote distinct primes. N denotes
the positive integers.

OEIS | formula | OEIS description | first 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 |

`A018252` |
N \ {p} |
The nonprime numbers (1 together with the composite numbers, A002808). | 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35 |

`A008578` |
{p, 1} |
Prime numbers at the beginning of the 20th century (today 1 is no longer regarded as a prime). | 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83 |

`A002808` |
N \ {p, 1} |
The composite numbers: numbers n of the form x*y for x > 1 and y > 1. | 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36 |

`A001248` |
{p^{2}} |
Squares of primes. | 4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721 |

`A280076` |
{p^{2}, 1} |
Numbers n such that Sum_{d|n} tau(d) = Product_{d|n} tau(d). | 1, 4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481 |

`A000430` |
{p^{2}, p} |
Primes and squares of primes. | 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71 |

`A030078` |
{p^{3}} |
Cubes of primes. | 8, 27, 125, 343, 1331, 2197, 4913, 6859, 12167, 24389, 29791, 50653, 68921, 79507, 103823 |

`A087797` |
{p^{3}, p^{2}, p} |
Primes, squares of primes and cubes of primes. | 2, 3, 4, 5, 7, 8, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61 |

`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 |

`A167171` |
{pq, p} |
Squarefree semiprimes together with primes. | 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39 |

`A001358` |
{pq, p^{2}} |
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 |

`A037143` |
{pq, p^{2}, p, 1} |
Numbers with at most 2 prime factors (counted with multiplicity). | 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35 |

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

`A285508` |
{p^{2}q, p^{3}} |
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 |

`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 |

`A217856` |
{pqr, p^{2}q} |
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` |
{pqr, p^{2}q, p^{3}} |
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 |

`A038609` |
{p+q} |
Numbers that are the sum of 2 different primes. | 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 28, 30, 31, 32, 33 |

`A166081` |
N \ {p+q} |
Natural numbers that not are the sum of two distinct primes. | 1, 2, 3, 4, 6, 11, 17, 23, 27, 29, 35, 37, 41, 47, 51, 53, 57, 59, 65, 67, 71, 77, 79, 83 |

`A014091` |
{p+q, 2p} |
Numbers that are the sum of 2 primes. | 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 28, 30, 31, 32 |

`A014092` |
N \ {p+q, 2p} |
Numbers that are not the sum of 2 primes. | 1, 2, 3, 11, 17, 23, 27, 29, 35, 37, 41, 47, 51, 53, 57, 59, 65, 67, 71, 77, 79, 83, 87 |

(My proposal for {`p``q`, `p`^{2}, `p`} was rejected for being too similar to A037143.)