Editing Super-Poulet number

{{Short description|Type of Poulet number}}
In [[number theory]], a '''super-Poulet number''' is a [[Poulet number]], or [[pseudoprime]] to [[Radix|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 [[Composite number|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 &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 factor]]s 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.

== External links ==
* {{mathworld|Super-PouletNumber}}
*[http://www.numericana.com/answer/pseudo.htm#super Numericana]
{{Classes of natural numbers}}

[[Category:Integer sequences]]