# Lucas-Carmichael number

Given an odd squarefree^{} integer $n$ (that is, one with factorization $n={\displaystyle \prod _{i=1}^{\omega (n)}}{p}_{i}$, 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 |
---|---|

Canonical name | LucasCarmichaelNumber |

Date of creation | 2013-03-22 17:41:14 |

Last modified on | 2013-03-22 17:41:14 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 6 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11A51 |