NationMaster - Encyclopedia: Almost prime

A natural number is thus prime iff it is 1-almost prime, and semiprime iff it is 2-almost prime. The set of k-almost prime numbers is usually denoted by P_k. 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 (http://www.research.att.com/projects/OEIS?Anum=A000040)
2	4, 6, 9, 10, 14, 15, 21, 22, ...	A001358 (http://www.research.att.com/projects/OEIS?Anum=A001358)
3	8, 12, 18, 20, 27, 28, 30, ...	A014612 (http://www.research.att.com/projects/OEIS?Anum=A014612)
4	16, 24, 36, 40, 54, 56, 60, ...	A014613 (http://www.research.att.com/projects/OEIS?Anum=A014613)
5	32, 48, 72, 80, 108, 112, ...	A014614 (http://www.research.att.com/projects/OEIS?Anum=A014614)
6	64, 96, 144, 160, 216, 224, ...	A046306 (http://www.research.att.com/projects/OEIS?Anum=A046306)
7	128, 192, 288, 320, 432, 448, ...	A046308 (http://www.research.att.com/projects/OEIS?Anum=A046308)
8	256, 384, 576, 640, 864, 896, ...	A046310 (http://www.research.att.com/projects/OEIS?Anum=A046310)
9	512, 768, 1152, 1280, 1728, ...	A046312 (http://www.research.att.com/projects/OEIS?Anum=A046312)
10	1024, 1536, 2304, 2560, ...	A046314 (http://www.research.att.com/projects/OEIS?Anum=A046314)
11	2048, 3072, 4608, 5120, ...	A069272 (http://www.research.att.com/projects/OEIS?Anum=A069272)
12	4096, 6144, 9216, 10240, ...	A069273 (http://www.research.att.com/projects/OEIS?Anum=A069273)
13	8192, 12288, 18432, 20480, ...	A069274 (http://www.research.att.com/projects/OEIS?Anum=A069274)
14	16384, 24576, 36864, 40960, ...	A069275 (http://www.research.att.com/projects/OEIS?Anum=A069275)
15	32768, 49152, 73728, 81920, ...	A069276 (http://www.research.att.com/projects/OEIS?Anum=A069276)
16	65536, 98304, 147456, ...	A069277 (http://www.research.att.com/projects/OEIS?Anum=A069277)
17	131072, 196608, 294912, ...	A069278 (http://www.research.att.com/projects/OEIS?Anum=A069278)
18	262144, 393216, 589824, ...	A069279 (http://www.research.att.com/projects/OEIS?Anum=A069279)
19	524288, 786432, 1179648, ...	A069280 (http://www.research.att.com/projects/OEIS?Anum=A069280)
20	1048576, 1572864, 2359296, ...	A069281 (http://www.research.att.com/projects/OEIS?Anum=A069281)

External links

MathWorld: Almost prime (http://mathworld.wolfram.com/AlmostPrime.html)

Results from FactBites:

Sieve theory - Wikipedia, the free encyclopedia (587 words)
	In one of the major strands of number theory in the twentieth century, ways were found of avoiding some of the difficulties of a frontal attack with a naive idea of what sieving should be.
	the set of almost prime numbers), which is typically somewhat larger than the original set, and easier to analyze.
	One of the original purposes of sieve theory was to try to prove conjectures in number theory such as the twin prime conjecture.

Puzzle 196. Consecutive integers with the same quantity of prime factors (1024 words)
	Note a number with exactly k prime factors, counting repetitions, is called a k-almost prime.
	I first computed sets of probable primes around 338 digits and then tried different combinations of these for a, b and c until the cofactors (x+1)/14 and (x+2)/15 were both probable primes.
	The probability of a d-digit number being prime is O(d).

More results at FactBites »

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