Lucas-Carmichael number

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In mathematics, a Lucas-Carmichael number is a positive integer n such that if p is a prime factor of n, then p + 1 is a factor of n + 1. By custom, 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

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

76751 = 23 × 47 × 71

80189 = 17 × 53 × 89

81719 = 11 × 17 × 19 × 23

88559 = 19 × 59 × 79

104663 = 13 × 83 × 97

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.