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殆素数

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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素因子[1][2]。自然数n称为k次殆素数当且仅当Ω(n) = k,其中Ω(n)是n整数分解过程中的指数和:

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

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Pk。开始的几个k次殆素数是:

k k次殆素数 OEIS数列
1 2, 3, 5, 7, 11, 13, 17, 19, ... OEISA000040
2 4, 6, 9, 10, 14, 15, 21, 22, ... OEISA001358
3 8, 12, 18, 20, 27, 28, 30, ... OEISA014612
4 16, 24, 36, 40, 54, 56, 60, ... OEISA014613
5 32, 48, 72, 80, 108, 112, ... OEISA014614
6 64, 96, 144, 160, 216, 224, ... OEISA046306
7 128, 192, 288, 320, 432, 448, ... OEISA046308
8 256, 384, 576, 640, 864, 896, ... OEISA046310
9 512, 768, 1152, 1280, 1728, ... OEISA046312
10 1024, 1536, 2304, 2560, ... OEISA046314
11 2048, 3072, 4608, 5120, ... OEISA069272
12 4096, 6144, 9216, 10240, ... OEISA069273
13 8192, 12288, 18432, 20480, ... OEISA069274
14 16384, 24576, 36864, 40960, ... OEISA069275
15 32768, 49152, 73728, 81920, ... OEISA069276
16 65536, 98304, 147456, ... OEISA069277
17 131072, 196608, 294912, ... OEISA069278
18 262144, 393216, 589824, ... OEISA069279
19 524288, 786432, 1179648, ... OEISA069280
20 1048576, 1572864, 2359296, ... OEISA069281

参考资料[编辑]

  1. ^ Sándor, József; Dragoslav, Mitrinović S.; Crstici, Borislav. Handbook of Number Theory I. Springer. 2006: 316. ISBN 978-1-4020-4215-7 (英文). 
  2. ^ Rényi, Alfréd A. On the representation of an even number as the sum of a single prime and single almost-prime number. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 1948, 12 (1): 57–78 (俄文). 

外部連結[编辑]