Lucas–Carmichael number

In mathematics, a Lucas–Carmichael number is a positive composite integer n such that if p is a prime factor of n, then p + 1 is a factor of n + 1. They are named after Édouard Lucas and Robert Carmichael.

By convention, a number is only regarded as a Lucas–Carmichael number if it is odd and square-free (not divisible by the square of a prime number), otherwise any cube of a prime number, such as 8 or 27, would be a Lucas–Carmichael number (since n³ + 1 = (n + 1)(n² − n + 1) is always divisible by n + 1).

Thus the smallest such number is 399 = 3 × 7 × 19; 399+1 = 400; 3+1, 7+1 and 19+1 are all factors of 400.

The first few numbers, and their factors, are (sequence A006972 in OEIS):

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

The smallest Lucas-Carmichael number with 5 factors is 588455 = 5 × 7 × 17 × 23 × 43.

It is not known whether any Lucas–Carmichael number is also a Carmichael number.
References

Unsolved Problems in Number Theory (3rd edition) by Richard Guy (Springer Verlag, 2004), section A13.
PlanetMath

Mathematics Encyclopedia