查看“︁殆素数”︁的源代码

{{Expand|time=2013-02-14T05:26:04+00:00}}
{{expert|time=2010-03-05T02:16:58+00:00|subject=数学}}
{{noteTA|G1=Math}}
[[数论]]中，一个[[自然数]]称为'''殆素数'''，[[当且仅当]]存在一个绝对常数''K''，使这个自然数最多有''K''个[[質因數|素因子]]<ref>{{cite book | last1 = Sándor | first1 = József | last2 = Dragoslav | first2 = Mitrinović S. | last3 = Crstici | first3 = Borislav | title = Handbook of Number Theory I | publisher = [[Springer Science+Business Media|Springer]] | year = 2006 | page = 316 | language = en | url = http://link.springer.com/referencework/10.1007%2F1-4020-3658-2 | isbn = 978-1-4020-4215-7 | access-date = 2015-04-14 | archive-date = 2021-03-08 | archive-url = https://web.archive.org/web/20210308072121/http://link.springer.com/referencework/10.1007/1-4020-3658-2 | dead-url = no }}</ref><ref>{{cite journal | title = On the representation of an even number as the sum of a single prime and single almost-prime number | first = Alfréd A. | last = Rényi | journal = Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya | volume = 12 | issue = 1 | year = 1948 | pages = 57–78 | language = ru | url = http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=3018&option_lang=eng | access-date = 2015-04-14 | archive-date = 2021-04-08 | archive-url = https://web.archive.org/web/20210408231103/http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=3018&option_lang=eng | dead-url = no }}</ref>。自然数''n''称为'''k次殆素数'''，[[当且仅当]]Ω(''n'') = ''k''，其中Ω（''n''）是''n''的[[整数分解]]过程中的指数和：

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

因此，一个自然数是[[素数]]，当且仅当它是一次殆素数；一个自然数是[[半素数]]，当且仅当它是二次殆素数。k次殆素数的集合通常表示成''P''<sub>''k''</sub>。开始的几个k次殆素数是：
:
{| class="wikitable"
|-----
! ''k''
! ''k''次殆素数
! [[OEIS]]数列
|-----
| 1 || 2, 3, 5, 7, 11, 13, 17, 19, ...
| {{oeis|A000040}}
|-----
| 2 || 4, 6, 9, 10, 14, 15, 21, 22, ...
| {{oeis|A001358}}
|-----
| 3 || 8, 12, 18, 20, 27, 28, 30, ...
| {{oeis|A014612}}
|-----
| 4 || 16, 24, 36, 40, 54, 56, 60, ...
| {{oeis|A014613}}
|-----
| 5 || 32, 48, 72, 80, 108, 112, ...
| {{oeis|A014614}}
|-----
| 6 || 64, 96, 144, 160, 216, 224, ...
| {{oeis|A046306}}
|-----
| 7 || 128, 192, 288, 320, 432, 448, ...
| {{oeis|A046308}}
|-----
| 8 || 256, 384, 576, 640, 864, 896, ...
| {{oeis|A046310}}
|-----
| 9 || 512, 768, 1152, 1280, 1728, ...
| {{oeis|A046312}}
|-----
| 10 || 1024, 1536, 2304, 2560, ...
| {{oeis|A046314}}
|-----
| 11 || 2048, 3072, 4608, 5120, ...
| {{oeis|A069272}}
|-----
| 12 || 4096, 6144, 9216, 10240,  ...
| {{oeis|A069273}}
|-----
| 13 || 8192, 12288, 18432, 20480, ...
| {{oeis|A069274}}
|-----
| 14 || 16384, 24576, 36864, 40960, ...
| {{oeis|A069275}}
|-----
| 15 || 32768, 49152, 73728, 81920, ...
| {{oeis|A069276}}
|-----
| 16 || 65536, 98304, 147456, ...
| {{oeis|A069277}}
|-----
| 17 || 131072, 196608, 294912, ...
| {{oeis|A069278}}
|-----
| 18 || 262144, 393216, 589824, ...
| {{oeis|A069279}}
|-----
| 19 || 524288, 786432, 1179648, ...
| {{oeis|A069280}}
|-----
| 20 || 1048576, 1572864, 2359296, ...
| {{oeis|A069281}}
|}

==参考资料==
{{Reflist}}

== 外部連結 ==
* [http://mathworld.wolfram.com/AlmostPrime.html MathWorld: Almost prime] {{Wayback|url=http://mathworld.wolfram.com/AlmostPrime.html |date=20210323023027 }}

{{質數}}
[[Category:整数数列|D]]