# 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;
* n is odd and square-free.
The first condition resembles Korselt's criterion for Carmichael numbers, where -1 is replaced with +1. The second condition eliminates from consideration some trivial cases like cubes of prime numbers, such as 8 or 27, which otherwise would be Lucas–Carmichael numbers (since n$^3$ + 1 = (n + 1)(n$^2$ − n + 1) is always divisible by n + 1).
They are named after Édouard Lucas and Robert Carmichael.

## Properties

The smallest Lucas–Carmichael number is 399 = 3 × 7 × 19. It is easy to verify that 3+1, 7+1, and 19+1 are all factors of 399+1 = 400.
The smallest Lucas–Carmichael number with 4 factors is 8855 = 5 × 7 × 11 × 23.
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.
Thomas Wright proved in 2016 that there are infinitely many Lucas–Carmichael numbers. If we let $N(X)$ denote the number of Lucas–Carmichael numbers up to $X$, Wright showed that there exists a positive constant $K$ such that
$N(X)\gg X^{K/\left(\log \log \log X\right)^{2}}$.

## List of Lucas–Carmichael numbers

The first few Lucas–Carmichael numbers (sequence A006972 in the OEIS) and their prime factors are listed below.
| 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 × 29 |
| 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 |