Almost prime

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):

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

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
12, 3, 5, 7, 11, 13, 17, 19, … A000040
24, 6, 9, 10, 14, 15, 21, 22, … A001358
38, 12, 18, 20, 27, 28, 30, … A014612
416, 24, 36, 40, 54, 56, 60, … A014613
532, 48, 72, 80, 108, 112, … A014614
664, 96, 144, 160, 216, 224, … A046306
7128, 192, 288, 320, 432, 448, … A046308
8256, 384, 576, 640, 864, 896, … A046310
9512, 768, 1152, 1280, 1728, … A046312
101024, 1536, 2304, 2560, … A046314
112048, 3072, 4608, 5120, … A069272
124096, 6144, 9216, 10240, … A069273
138192, 12288, 18432, 20480, … A069274
1416384, 24576, 36864, 40960, … A069275
1532768, 49152, 73728, 81920, … A069276
1665536, 98304, 147456, … A069277
17131072, 196608, 294912, … A069278
18262144, 393216, 589824, … A069279
19524288, 786432, 1179648, … A069280
201048576, 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]

a result of Landau.[5] See also the Hardy–Ramanujan theorem.

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". Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya (in Russian). 12 (1): 57–78.
  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. Bibcode:1978MPCPS..83..357H. doi:10.1017/S0305004100054657.
  4. Tenenbaum, Gerald (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge University Press. ISBN 978-0-521-41261-2.
  5. Landau, Edmund (1953) [first published 1909]. "§ 56, Über Summen der Gestalt ". Handbuch der Lehre von der Verteilung der Primzahlen. vol. 1. Chelsea Publishing Company. p. 211.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.