A super-Poulet number is a Poulet number whose every divisor

Divisor

In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...

 d divides

2^d − 2.

For example 341 is a super-Poulet number: it has positive divisors {1, 11, 31, 341} and we have: / 11 = 2046 / 11 = 186 / 31 = 2147483646 / 31 = 69273666 / 341 = 13136332798696798888899954724741608669335164206654835981818117894215788100763407304286671514789484550
The super-Poulet numbers below 10,000 are :

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

Super-Poulet numbers with 3 or more distinct prime divisors

It is relatively easy to get super-Poulet numbers with 3 distinct prime divisors. If you find three Poulet numbers with three common prime factors, you get a super-Poulet number, as you built the product of the three prime factors.

Example:
2701 = 37 * 73 is a Poulet number
4033 = 37 * 109 is a Poulet number
7957 = 73 * 109 is a Poulet number

so 294409 = 37 * 73 * 109 is a Poulet number too.

Super-Poulet numbers with up to 7 distinct prime factor

Prime factor

In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly, without leaving a remainder. The process of finding these numbers is called integer factorization, or prime factorization. A prime factor can be visualized by understanding Euclid's...

s you can get with the following numbers:

• { 103, 307, 2143, 2857, 6529, 11119, 131071 }
• { 709, 2833, 3541, 12037, 31153, 174877, 184081 }
• { 1861, 5581, 11161, 26041, 37201, 87421, 102301 }
• { 6421, 12841, 51361, 57781, 115561, 192601, 205441 }

For example 1.118.863.200.025.063.181.061.994.266.818.401 = 6421 * 12841 * 51361 * 57781 * 115561 * 192601 * 205441 is a super-Poulet number with 7 distinct prime factors and 120 Poulet numbers.