In number theory, a super-Poulet number is a Poulet number, or pseudoprime to base 2, whose every divisor <math>d</math> divides <math>2^d - 2</math>.

For example, 341 is a super-Poulet number: it has positive divisors (1, 11, 31, 341), and we have:

:(2<sup>11</sup> − 2) / 11 = 2046 / 11 = 186

:(2<sup>31</sup> − 2) / 31 = 2147483646 / 31 = 69273666

:(2<sup>341</sup> − 2) / 341 = 13136332798696798888899954724741608669335164206654835981818117894215788100763407304286671514789484550

When <math> \frac{ \Phi_n(2)}{gcd(n, \Phi_n(2))}</math> 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 :

{| class="wikitable"

|-

!n

!

|-

| 1

| 341 = 11 &times; 31

|-

| 2

| 1387 = 19 &times; 73

|-

| 3

| 2047 = 23 &times; 89

|-

| 4

| 2701 = 37 &times; 73

|-

| 5

| 3277 = 29 &times; 113

|-

| 6

| 4033 = 37 &times; 109

|-

| 7

| 4369 = 17 &times; 257

|-

| 8

| 4681 = 31 &times; 151

|-

| 9

| 5461 = 43 &times; 127

|-

| 10

| 7957 = 73 &times; 109

|-

| 11

| 8321 = 53 &times; 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:

<!-- from http://www.numericana.com/answer/pseudo.htm#poulet, from Gerard Michon -->

  • { 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.

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