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# 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.

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## Mathematics Subject Classification

11A51*no label found*

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## Comments

## Extremely minor detail

I wouldn't want to file this as a correction, I'd just suggest saying, after explaining what omega(n) is, that no p_i = 2 (as earlier you said n is odd here).

## Re: Extremely minor detail

Thanks for the suggestion.