A super-Poulet number is a Poulet number whose every divisor d divides 2d - 2. For example 341 is a super-Poulet number: it has positive divisors {1, 11, 31, 341} and we have:
(211 - 2) / 11 = 2046 / 11 = 186
(231 - 2) / 31 = 2147483646 / 31 = 69273666
(2341 - 2) / 341 = ... (an integer)

The super-Poulet numbers below 10000 are (SIDN A050217):

n 
1 341 = 11 × 31
2 1387 = 19 × 73
3 2047 = 23 × 89
4 2701 = 37 × 73
5 3277 = 29 × 112
6 4033 = 37 × 109
7 4369 = 17 × 257
8 4681 = 31 × 151
9 5461 = 43 × 127
10 7957 = 73 × 109
11 8321 = 53 × 157