# Super-Poulet number

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

When a composite number is a pseudoprime to base 2, but not to every base (That is, not a Carmichael number), then it is a super-Poulet number, and 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 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

## 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 factors 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.