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Almost prime

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Almost prime

In number theory, a natural number is called almost prime if there exists an absolute constant K such that the number has at most K prime factors.[1][2] An almost prime n is denoted by Pr if and only if the number of prime factors of n, counted according to multiplicity, is at most r.[3] A natural number is called k-almost prime 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 total number of primes in the prime factorization of n:

\Omega(n) := \sum a_i \qquad\mbox{if}\qquad n = \prod p_i^{a_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 unique 0-almost prime is 1 (i.e., the empty product). By the fundamental theorem of arithmetic, the set of all sets of k-almost primes for all k \in \{0,1,2,3,...\} constitutes a partition of the positive integers.

The first few k-almost primes are:

k k-almost primes OEIS sequence
0 1
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

The number πk(n) of positive integers less than or equal to n with at most k prime divisors (not necessarily distinct) is asymptotic to:[4]

\pi_k(n) \sim \left( \frac{n}{\log n} \right) \frac{(\log\log n)^{k-1}}{(k - 1)!},

a result of Landau. See also the Hardy–Ramanujan theorem.

References

  1. ^ Sándor, József; Dragoslav, Mitrinović S.; Crstici, Borislav (2006). Handbook of Number Theory I.  
  2. ^  
  3. ^  
  4. ^  

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