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 （OEIS數列A006972）:
| --- | --- |
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.