en>Yobot: WP:CHECKWIKI error fixes using AWB (10093)

2014-05-05T12:04:59Z

WP:CHECKWIKI error fixes using AWB (10093)

en>Myasuda: /* Historical figures */ added missing diacritics

2014-01-31T02:54:06Z

Historical figures: added missing diacritics

← Older revision		Revision as of 04:54, 31 January 2014
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			In [[number theory]], a [[natural number]] is called '''''k''-almost prime''' if it has exactly ''k'' [[prime factor]]s, counted with multiplicity. More formally, a number ''n'' is ''k''-almost prime if and only if [[arithmetic function#Ω(n), ω(n), νp(n) – prime power decomposition\|Ω]](''n'') = ''k'', where Ω(''n'') is the total number of primes in the [[prime factorization]] of ''n'':

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

	~~Aleta~~ is ~~what~~'~~s written~~ by ~~her birth certificate nevertheless she doesn~~'~~t really these being called like a~~. ~~Massachusetts is often where he~~'~~s always lived. Managing men~~ is ~~what she definitely does in her~~ [~~http:~~/~~/Daywriting~~.~~com/ day writing] but she~~'~~s always wished her own business. To motivate is something her husband doesn~~'~~t really like but yet she does. She is [http~~:~~//Www.Reddit.com/r/howto/search?q~~=~~running running~~] ~~and consistently a blog here: http://circuspartypanama.com~~<br><br>~~Feel free~~ to ~~surf~~ to ~~my blog post~~: [~~http~~://~~circuspartypanama.com clash~~ of ~~clans tricks~~]		A natural number is thus [[prime number\|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''<sub>''k''</sub>. The smallest ''k''-almost prime is [[Power of two\|2<sup>''k''</sup>]]. The first few ''k''-almost primes are:

			:
			{\| class="wikitable"
			\|-
			! ''k''
			! ''k''-almost primes
			! [[OEIS]] sequence
			\|-
			\| 1 \|\| 2, 3, 5, 7, 11, 13, 17, 19, …
			\| {{OEIS link\|id=A000040}}
			\|-
			\| 2 \|\| 4, 6, 9, 10, 14, 15, 21, 22, …
			\| {{OEIS link\|id=A001358}}
			\|-
			\| 3 \|\| 8, 12, 18, 20, 27, 28, 30, …
			\| {{OEIS link\|id=A014612}}
			\|-
			\| 4 \|\| 16, 24, 36, 40, 54, 56, 60, …
			\| {{OEIS link\|id=A014613}}
			\|-
			\| 5 \|\| 32, 48, 72, 80, 108, 112, …
			\| {{OEIS link\|id=A014614}}
			\|-
			\| 6 \|\| 64, 96, 144, 160, 216, 224, …
			\| {{OEIS link\|id=A046306}}
			\|-
			\| 7 \|\| 128, 192, 288, 320, 432, 448, …
			\| {{OEIS link\|id=A046308}}
			\|-
			\| 8 \|\| 256, 384, 576, 640, 864, 896, …
			\| {{OEIS link\|id=A046310}}
			\|-
			\| 9 \|\| 512, 768, 1152, 1280, 1728, …
			\| {{OEIS link\|id=A046312}}
			\|-
			\| 10 \|\| 1024, 1536, 2304, 2560, …
			\| {{OEIS link\|id=A046314}}
			\|-
			\| 11 \|\| 2048, 3072, 4608, 5120, …
			\| {{OEIS link\|id=A069272}}
			\|-
			\| 12 \|\| 4096, 6144, 9216, 10240, …
			\| {{OEIS link\|id=A069273}}
			\|-
			\| 13 \|\| 8192, 12288, 18432, 20480, …
			\| {{OEIS link\|id=A069274}}
			\|-
			\| 14 \|\| 16384, 24576, 36864, 40960, …
			\| {{OEIS link\|id=A069275}}
			\|-
			\| 15 \|\| 32768, 49152, 73728, 81920, …
			\| {{OEIS link\|id=A069276}}
			\|-
			\| 16 \|\| 65536, 98304, 147456, …
			\| {{OEIS link\|id=A069277}}
			\|-
			\| 17 \|\| 131072, 196608, 294912, …
			\| {{OEIS link\|id=A069278}}
			\|-
			\| 18 \|\| 262144, 393216, 589824, …
			\| {{OEIS link\|id=A069279}}
			\|-
			\| 19 \|\| 524288, 786432, 1179648, …
			\| {{OEIS link\|id=A069280}}
			\|-
			\| 20 \|\| 1048576, 1572864, 2359296, …
			\| {{OEIS link\|id=A069281}}
			\|}

			The number π<sub>''k''</sub>(''n'') of positive integers less than or equal to ''n'' with at most ''k'' prime divisors (not necessarily distinct) is [[asymptotic analysis\|asymptotic]] to:<ref>{{cite book
			\| last = Tenenbaum
			\| first = Gerald
			\| authorlink = Gerald Tenenbaum
			\| coauthors =
			\| title = Introduction to Analytic and Probabilistic Number Theory
			\| publisher = [[Cambridge University Press]]
			\| date = 1995
			\| isbn = 0-521-41261-7}}</ref>

			: <math> \pi_k(n) \sim \left( \frac{n}{\log n} \right) \frac{(\log\log n)^{k-1}}{(k - 1)!},</math>
			a result of [[Edmund Landau\|Landau]]. See also the [[Hardy–Ramanujan theorem]].

			==References==
			<references/>

			== External links ==
			* {{MathWorld\|urlname=AlmostPrime\|title=Almost prime}}

			{{Prime number classes}}
			{{Classes of natural numbers}}

			[[Category:Integer sequences]]
			[[Category:Prime numbers]]

en>EmausBot: r2.7.2+) (Robot: Modifying ar:ملحق:قائمة المواضيع المتعلقة بالتحليل الحقيقي

2012-07-07T19:22:39Z

r2.7.2+) (Robot: Modifying ar:ملحق:قائمة المواضيع المتعلقة بالتحليل الحقيقي

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Aleta is what's written by her birth certificate nevertheless she doesn't really these being called like a. Massachusetts is often where he's always lived. Managing men is what she definitely does in her [http://Daywriting.com/ day writing] but she's always wished her own business. To motivate is something her husband doesn't really like but yet she does. She is [http://Www.Reddit.com/r/howto/search?q=running running] and consistently a blog here: http://circuspartypanama.com<br><br>Feel free to surf to my blog post: [http://circuspartypanama.com clash of clans tricks]

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en>Yobot: WP:CHECKWIKI error fixes using AWB (10093)

en>Myasuda: /* Historical figures */ added missing diacritics

en>EmausBot: r2.7.2+) (Robot: Modifying ar:ملحق:قائمة المواضيع المتعلقة بالتحليل الحقيقي