# super-Poulet number

A $n$ is a Poulet number which besides satisfying the congruence $2^{n}\equiv 2\mod n$, each of its divisors $d_{i}$ (for $1) also satisfies the congruence $2^{d_{i}}\equiv 2\mod d_{i}$.

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: $\frac{2^{33}-2}{33}\approx 260301048.18181818\ldots$.

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 SuperPouletNumber 2013-03-22 18:14:12 2013-03-22 18:14:12 PrimeFan (13766) PrimeFan (13766) 4 PrimeFan (13766) Definition msc 11A51