# super-Poulet number

A super-Poulet number^{} $n$ is a Poulet number^{} which besides satisfying the congruence^{} ${2}^{n}\equiv 2modn$, each of its divisors^{} ${d}_{i}$ (for $$) also satisfies the congruence ${2}^{{d}_{i}}\equiv 2mod{d}_{i}$.

Two examples: 341 is a super-Poulet number, with its divisors being 1, 11, 31 and 341 itself. We verify that ${2}^{11}=2048=11\times 186+2$ and ${2}^{31}=2147483648=31\times 69273666+2$. 341 itself has already been checked when confirmed as a Poulet number. Now, 561 is a Poulet number but not a super-Poulet number since one of its divisors, 33, does not satisfy the congruence: $\frac{{2}^{33}-2}{33}\approx 260301048.18181818\mathrm{\dots}$.

The first few super-Poulet numbers are 341, 1387, 2047, 2701, 3277, 4033, 4369, 4681, 5461, 7957, 8321, which are listed in A050217 of Sloane’s OEIS.

Title | super-Poulet number |
---|---|

Canonical name | SuperPouletNumber |

Date of creation | 2013-03-22 18:14:12 |

Last modified on | 2013-03-22 18:14:12 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 4 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11A51 |