Almost prime

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Demonstration, with Cuisenaire rods, of the 2-almost prime nature of the number 6

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):

[math]\displaystyle{ \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 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

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]Template:Relevance?

[math]\displaystyle{ \pi_k(n) \sim \left( \frac{n}{\log n} \right) \frac{(\log\log n)^{k-1}}{(k - 1)!}, }[/math]

a result of Landau.[5] See also the Hardy–Ramanujan theorem.Template:Relevance?

Properties

  • The multiple of a [math]\displaystyle{ k_1 }[/math]-almost prime and a [math]\displaystyle{ k_2 }[/math]-almost prime is a [math]\displaystyle{ (k_1+k_2) }[/math]-almost prime.
  • A [math]\displaystyle{ k }[/math]-almost prime cannot have a [math]\displaystyle{ n }[/math]-almost prime as a factor for all [math]\displaystyle{ n\gt k }[/math].

References

  1. ↑ Sándor, József; Dragoslav, Mitrinović S.; Crstici, Borislav (2006). Handbook of Number Theory I. Springer. p. 316. doi:10.1007/1-4020-3658-2. ISBN 978-1-4020-4215-7. 
  2. ↑ Rényi, Alfréd A. (1948). "On the representation of an even number as the sum of a single prime and single almost-prime number" (in ru). Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 12 (1): 57–78. http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=3018&option_lang=eng. 
  3. ↑ Heath-Brown, D. R. (May 1978). "Almost-primes in arithmetic progressions and short intervals". Mathematical Proceedings of the Cambridge Philosophical Society 83 (3): 357–375. doi:10.1017/S0305004100054657. Bibcode: 1978MPCPS..83..357H. 
  4. ↑ Tenenbaum, Gerald (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge University Press. ISBN 978-0-521-41261-2. 
  5. ↑ Landau, Edmund (1953). "§ 56, Über Summen der Gestalt [math]\displaystyle{ \sum_{p\leq x}F(p,x) }[/math]". Handbuch der Lehre von der Verteilung der Primzahlen. 1. Chelsea Publishing Company. p. 211. 

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