Almost prime
From Wikipedia, the free encyclopedia
In mathematics, a natural number is called k-almost prime if and only if it has exactly k prime factors, counted with multiplicity. More formally, a number n is k-almost prime if and only if Ω(n) = k, where Ω(n) is the sum of the exponents in the prime factorization of n:
A natural number is thus prime if and only if it is 1-almost prime, and semiprime if and only if it is 2-almost prime. The set of k-almost primes is usually denoted by Pk. The smallest k-almost prime is 2k. The first few k-almost primes are:
| k | k-almost primes | OEIS sequence |
|---|---|---|
| 1 | 2, 3, 5, 7, 11, 13, 17, 19, ... | A000040 |
| 2 | 4, 6, 9, 10, 14, 15, 21, 22, ... | A001358 |
| 3 | 8, 12, 18, 20, 27, 28, 30, ... | A014612 |
| 4 | 16, 24, 36, 40, 54, 56, 60, ... | A014613 |
| 5 | 32, 48, 72, 80, 108, 112, ... | A014614 |
| 6 | 64, 96, 144, 160, 216, 224, ... | A046306 |
| 7 | 128, 192, 288, 320, 432, 448, ... | A046308 |
| 8 | 256, 384, 576, 640, 864, 896, ... | A046310 |
| 9 | 512, 768, 1152, 1280, 1728, ... | A046312 |
| 10 | 1024, 1536, 2304, 2560, ... | A046314 |
| 11 | 2048, 3072, 4608, 5120, ... | A069272 |
| 12 | 4096, 6144, 9216, 10240, ... | A069273 |
| 13 | 8192, 12288, 18432, 20480, ... | A069274 |
| 14 | 16384, 24576, 36864, 40960, ... | A069275 |
| 15 | 32768, 49152, 73728, 81920, ... | A069276 |
| 16 | 65536, 98304, 147456, ... | A069277 |
| 17 | 131072, 196608, 294912, ... | A069278 |
| 18 | 262144, 393216, 589824, ... | A069279 |
| 19 | 524288, 786432, 1179648, ... | A069280 |
| 20 | 1048576, 1572864, 2359296, ... | A069281 |
[edit] External links
- Eric W. Weisstein, Almost prime at MathWorld.
