thumb|Demonstration, with [[Cuisenaire rods, of the 2-almost prime nature of the number 6]]
In number theory, a natural number is called -almost prime if it has prime factors. More formally, a number is -almost prime if and only if , where is the total number of primes in the prime factorization of (can be also seen as the sum of all the primes' exponents):
:<math>\Omega(n) := \sum a_i \qquad\mbox{if}\qquad n = \prod p_i^{a_i}.</math>
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 -almost primes is usually denoted by . The smallest -almost prime is . The first few -almost primes are:
{|class="wikitable"
|-
!
! -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, …
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|-
| 4 || 16, 24, 36, 40, 54, 56, 60, …
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|-
| 5 || 32, 48, 72, 80, 108, 112, …
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|-
| 6 || 64, 96, 144, 160, 216, 224, …
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|-
| 7 || 128, 192, 288, 320, 432, 448, …
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|-
| 8 || 256, 384, 576, 640, 864, 896, …
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|-
| 9 || 512, 768, 1152, 1280, 1728, …
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|-
| 10 || 1024, 1536, 2304, 2560, …
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|-
| 11 || 2048, 3072, 4608, 5120, …
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|-
| 12 || 4096, 6144, 9216, 10240, …
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|-
| 13 || 8192, 12288, 18432, 20480, …
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|-
| 14 || 16384, 24576, 36864, 40960, …
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|-
| 15 || 32768, 49152, 73728, 81920, …
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|-
| 16 || 65536, 98304, 147456, …
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|-
| 17 || 131072, 196608, 294912, …
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|-
| 18 || 262144, 393216, 589824, …
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|-
| 19 || 524288, 786432, 1179648, …
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| 20 || 1048576, 1572864, 2359296, …
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|}
The number of positive integers less than or equal to with exactly prime divisors (not necessarily distinct) is asymptotic to:
: <math> \pi_k(n) \sim \left( \frac{n}{\log n} \right) \frac{(\log\log n)^{k-1{(k - 1)!},</math>
a result of Landau. See also the Hardy–Ramanujan theorem.
Properties
- The product of a -almost prime and a -almost prime is a -almost prime.
- A -almost prime cannot have a -almost prime as a factor for all .
References
See also
- sphenic number – name for square-free 3-almost primes