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):
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][relevant?]
a result of Landau.[5] See also the Hardy–Ramanujan theorem.[relevant?]
^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.
^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.
^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. S2CID 122691474.
^Tenenbaum, Gerald (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge University Press. ISBN 978-0-521-41261-2.
^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.
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