In number theory, a natural number is called k-almost prime if it has k prime factors.[1] [2] [3] More formally, a number n is k-almost prime if and only if Ω(n) = k, where Ω(n) is the total number of primes in the prime factorization of n (can be also seen as the sum of all the primes' exponents):
\Omega(n):=\sumai if n=\prod
| ai | |
| p | |
| i |
.
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, … | ||
| 2 | 4, 6, 9, 10, 14, 15, 21, 22, … | ||
| 3 | 8, 12, 18, 20, 27, 28, 30, … | ||
| 4 | 16, 24, 36, 40, 54, 56, 60, … | ||
| 5 | 32, 48, 72, 80, 108, 112, … | ||
| 6 | 64, 96, 144, 160, 216, 224, … | ||
| 7 | 128, 192, 288, 320, 432, 448, … | ||
| 8 | 256, 384, 576, 640, 864, 896, … | ||
| 9 | 512, 768, 1152, 1280, 1728, … | ||
| 10 | 1024, 1536, 2304, 2560, … | ||
| 11 | 2048, 3072, 4608, 5120, … | ||
| 12 | 4096, 6144, 9216, 10240, … | ||
| 13 | 8192, 12288, 18432, 20480, … | ||
| 14 | 16384, 24576, 36864, 40960, … | ||
| 15 | 32768, 49152, 73728, 81920, … | ||
| 16 | 65536, 98304, 147456, … | ||
| 17 | 131072, 196608, 294912, … | ||
| 18 | 262144, 393216, 589824, … | ||
| 19 | 524288, 786432, 1179648, … | ||
| 20 | 1048576, 1572864, 2359296, … |
The number πk(n) of positive integers less than or equal to n with exactly k prime divisors (not necessarily distinct) is asymptotic to:[4]
\pik(n)\sim\left(
| n | |
| logn |
\right)
| (loglogn)k-1 | |
| (k-1)! |
,
k1
k2
(k1+k2)
k
n
n>k
\sump\leqF(p,x)