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):

${\displaystyle \Omega (n):=\sum a_{i}\qquad {\mbox{if}}\qquad n=\prod p_{i}^{a_{i}}.}$

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 P_k. The smallest k-almost prime is 2^k. 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 π_k(n) of positive integers less than or equal to n with exactly k prime divisors (not necessarily distinct) is asymptotic to:^{[4][relevant?]}

${\displaystyle \pi _{k}(n)\sim \left({\frac {n}{\log n}}\right){\frac {(\log \log n)^{k-1}}{(k-1)!}},}$

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

Properties

• The product of a k₁-almost prime and a k₂-almost prime is a (k₁ + k₂)-almost prime.
• A k-almost prime cannot have a n-almost prime as a factor for all n > k.

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. S2CID 122691474.
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 ${\displaystyle \sum _{p\leq x}F(p,x)}$". Handbuch der Lehre von der Verteilung der Primzahlen. Vol. 1. Chelsea Publishing Company. p. 211.

External links

• Weisstein, Eric W. "Almost prime". MathWorld.