Almost prime

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In mathematics, a natural number is called k-almost prime if and only 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 sum of the exponents in the prime factorization of n:

\Omega(n) := \sum a_i \qquad\mbox{if}\qquad n = \prod p_i^{a_i}.

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 prime numbers is usually denoted by Pk. The smallest k-almost prime is 2k. The first few k-almost prime numbers are:

k k-almost prime numbers 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

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