卢卡斯-卡米切尔数

在数学中，卢卡斯-卡米切尔数 是一个满足n的正合数，即如果 p 是 n 的质因子，那么 p + 1 是n + 1的因子。 而該数以 爱德华·卢卡斯 和 羅伯特·丹尼·卡邁克爾命名。

按照约定，一个被称作卢卡斯-卡米切尔数的数需在當且仅當它是奇数并是 无平方数因数的数 (不能被一个质数的平方整除)的情況下才能作此称謂，否则任何质数的立方，如8和27，都将成为卢卡斯-卡米切尔数 (因为 n3 + 1 = (n + 1)(n2 − n + 1) 一定可以被 n + 1整除)。

满足条件的最小的数是 399 = 3 × 7 × 19; 399+1 = 400; 3+1，7+1 和 19+1 都是400的因子。 

卢卡斯-卡米切尔数的前几个数及他们的因子是 :

 399    = 3 × 7 × 19 935    = 5 × 11 × 17 2015   = 5 × 13 × 31 2915   = 5 × 11 × 53 4991   = 7 × 23 × 31 5719   = 7 × 19 × 43 7055   = 5 × 17 × 83 8855   = 5 × 7 × 11 × 23 12719  = 7 × 23 × 79 18095  = 5 × 7 × 11 × 47 20705  = 5 × 41 × 101 20999  = 11 × 23 × 83 22847  = 11 × 31 × 67 29315  = 5 × 11 × 13 × 41 31535  = 5 × 7 × 17 × 53 46079  = 11 × 59 × 71 51359  = 7 × 11 × 23 × 291 60059  = 19 × 29 × 109 63503  = 11 × 23 × 251 67199  = 11 × 41 × 149 73535  = 5 × 7 × 11 × 191 76751  = 23 × 47 × 71 80189  = 17 × 53 × 89 81719  = 11 × 17 × 19 × 23 88559  = 19 × 59 × 79 90287  = 17 × 47 × 113 104663  = 13 × 83 × 97 117215  = 5 × 7 × 17 × 197 120581   = 17 × 41 × 173 147455   = 5 × 7 × 11 × 383 152279   = 29 × 59 × 89 155819   = 19 × 59 × 139 162687   = 3 × 7 × 61 × 127 191807   = 7 × 11 × 47 × 53 194327   = 7 × 17 × 23 × 71 196559   = 11 × 107 × 167 214199   = 23 × 67 × 139 218735   = 5 × 11 × 41 × 97 230159   = 47 × 59 × 83 265895   = 5 × 7 × 71 × 107 357599   = 11 × 19 × 29 × 59 388079   = 23 × 47 × 359 390335   = 5 × 11 × 47 × 151 482143   = 31 × 103 × 151 588455   = 5 × 7 × 17 × 23 × 43 653939   = 11 × 13 × 17 × 269 663679   = 31 × 79 × 271 676799   = 19 × 179 × 199 709019   = 17 × 179 × 233 741311   = 53 × 71 × 197 760655   = 5 × 7 × 103 × 211 761039   = 17 × 89 × 503 776567   = 11 × 227 × 311 798215   = 5 × 11 × 23 × 631 880319   = 11 × 191 × 419 895679   = 17 × 19 × 47 × 59 913031   = 7 × 23 × 53 × 107 966239   = 31 × 71 × 439 966779   = 11 × 179 × 491 973559   = 29 × 59 × 569 1010735  = 5 × 11 × 17 × 23 × 47 1017359  = 7 × 23 × 71 × 89 1097459  = 11 × 19 × 59 × 89 1162349  = 29 × 149 × 269 1241099  = 19 × 83 × 787 1256759  = 7 × 17 × 59 × 179 1525499  = 53 × 107 × 269 1554119  = 7 × 53 × 59 × 71 1584599  = 37 × 113 × 379 1587599  = 13 × 97 × 1259 1659119  = 7 × 11 × 29 × 743 1707839  = 7 × 29 × 47 × 179 1710863  = 7 × 11 × 17 × 1307 1719119  = 47 × 79 × 463 1811687  = 23 × 227 × 347 1901735  = 5 × 11 × 71 × 487 1915199  = 11 × 13 × 59 × 227 1965599  = 79 × 139 × 179 2048255  = 5 × 11 × 167 × 223 2055095  = 5 × 7 × 71 × 827 2150819  = 11 × 19 × 41 × 251 2193119  = 17 × 23 × 71 × 79 2249999  = 19 × 79 × 1499 2276351  = 7 × 11 × 17 × 37 × 47 2416679  = 23 × 179 × 587 2581319  = 13 × 29 × 41 × 167 2647679  = 31 × 223 × 383 2756159  = 7 × 17 × 19 × 23 × 53 2924099  = 29 × 59 × 1709 3106799  = 29 × 149 × 719 3228119  = 19 × 23 × 83 × 89 3235967  = 7 × 17 × 71 × 383 3332999  = 19 × 23 × 29 × 263 3354695  = 5 × 17 × 61 × 647 3419999  = 11 × 29 × 71 × 151 3441239  = 109 × 131 × 241 3479111  = 83 × 167 × 251 3483479  = 19 × 139 × 1319 3700619  = 13 × 41 × 53 × 131 3704399  = 47 × 269 × 293 3741479  = 7 × 17 × 23 × 1367 4107455  = 5 × 11 × 17 × 23 × 191 4285439  = 89 × 179 × 269 4452839  = 37 × 151 × 797 4587839  = 53 × 107 × 809 4681247  = 47 × 103 × 967 4853759  = 19 × 23 × 29 × 383 4874639  = 7 × 11 × 29 × 37 × 59 5058719  = 59 × 179 × 479 5455799  = 29 × 419 × 449 5669279  = 7 × 11 × 17 × 61 × 71 5807759  = 83 × 167 × 419 6023039  = 11 × 29 × 79 × 239 6514199  = 43 × 197 × 769 6539819  = 11 × 13 × 19 × 29 × 83 6656399  = 29 × 89 × 2579 6730559  = 11 × 23 × 37 × 719 6959699  = 59 × 179 × 659 6994259  = 17 × 467 × 881 7110179  = 37 × 41 × 43 × 109 7127999  = 23 × 479 × 647 7234163  = 17 × 41 × 97 × 107 7274249  = 17 × 449 × 953 7366463  = 13 × 23 × 71 × 347 8159759  = 19 × 29 × 59 × 251 8164079  = 7 × 11 × 229 × 463 8421335  = 5 × 13 × 23 × 43 × 131 8699459  = 43 × 307 × 659 8734109  = 37 × 113 × 2089 9224279  = 53 × 269 × 647 9349919  = 19 × 29 × 71 × 239 9486399  = 3 × 13 × 79 × 3079 9572639  = 29 × 41 × 83 × 97 9694079  = 47 × 239 × 863 9868715  = 5 × 43 × 197 × 233

最小的有5个因子的卢卡斯-卡米切尔数是 588455 = 5 × 7 × 17 × 23 × 43.

参考
 PlanetMath 

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