A super-Poulet number is a Poulet number whose every divisor d divides 2^d - 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) (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=050217):

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