A super-Poulet number $n$ is a Poulet number which besides satisfying the congruence $2^n \equiv 2 \mod n$ , each of its divisors $d_i$ (for $1 < i \leq \tau(n)$ ) 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.