Super-Poulet number

Super-Poulet number

A super-Poulet number is a Poulet number 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 = ... (an integer)

The super-Poulet numbers below 10,000 are OEIS|id=A050217:

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 number4033 = 37 * 109 is a Poulet number7957 = 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.

External links

*
* [http://home.att.net/~numericana/answer/pseudo.htm#poulet Numericana]


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