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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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## Recent Activity

Oct 21

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

## 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.