ALMOST PRIME

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'':
:$Omega(n)\; :=\; sum\; a\_i\; qquadmbox\{if\}qquad\; n\; =\; prod\; p\_i^\{a\_i\}.$
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 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

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