Super-Poulet number

In number theory, 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 {{OEIS|id=A050217}}:

class="wikitable"
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

class="wikitable"

| 341 = 11 × 31

| 1387 = 19 × 73

| 2047 = 23 × 89

| 2701 = 37 × 73

| 3277 = 29 × 113

| 4033 = 37 × 109

| 4369 = 17 × 257

| 4681 = 31 × 151

| 5461 = 43 × 127

| 7957 = 73 × 109

| 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, 1118863200025063181061994266818401 = 6421 * 12841 * 51361 * 57781 * 115561 * 192601 * 205441 is a super-Poulet number with 7 distinct prime factors and 120 Poulet numbers.

External links

{{mathworld|Super-PouletNumber}}
[https://www.numericana.com/answer/pseudo.htm#super Numericana]

Category:Integer sequences