Super-Poulet number

A super-Poulet number is a Poulet number, or pseudoprime to base 2, 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 = 13136332798696798888899954724741608669335164206654835981818117894215788100763407304286671514789484550

When ${\frac {\Phi _{n}(2)}{gcd(n,\Phi _{n}(2))}}$ is not prime, then it and every divisor of it are a pseudoprime to base 2, and a super-Poulet number.

The super-Poulet numbers below 10,000 are (sequence A050217 in the OEIS):

n
1	341 = 11 × 31
2	1387 = 19 × 73
3	2047 = 23 × 89
4	2701 = 37 × 73
5	3277 = 29 × 113
6	4033 = 37 × 109
7	4369 = 17 × 257
8	4681 = 31 × 151
9	5461 = 43 × 127
10	7957 = 73 × 109
11	8321 = 53 × 157