Almost prime

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 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:

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 at most k prime divisors (not necessarily distinct) is asymptotic to:[4]

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. Springer. p. 316. 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. doi:10.1017/S0305004100054657.
  4. Tenenbaum, Gerald (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge University Press. ISBN 0-521-41261-7.
  • Weisstein, Eric W. "Almost prime". MathWorld.
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