# Lucas-Carmichael number

Given an odd squarefree integer $n$ (that is, one with factorization $\displaystyle n=\prod_{i=1}^{\omega(n)}p_{i}$, with $\omega(n)$ being the number of distinct prime factors function, and all $p_{i}>2$) if it the case that each $p_{i}+1$ is a divisor of $n+1$, then $n$ is called a Lucas-Carmichael number.

For example, 935 has three prime factors, 5, 11, 17. Adding one to each of these we get 6, 12, 18, and these three numbers are all divisors of 936. Therefore, 935 is a Lucas-Carmichael number.

The first few Lucas-Carmichael numbers are 399, 935, 2015, 2915, 4991, 5719, 7055, 8855. These are listed in A006972 of Sloane’s OEIS.

Not to be confused with Carmichael numbers, the absolute Fermat pseudoprimes.

Title Lucas-Carmichael number LucasCarmichaelNumber 2013-03-22 17:41:14 2013-03-22 17:41:14 PrimeFan (13766) PrimeFan (13766) 6 PrimeFan (13766) Definition msc 11A51