Super-Poulet number
From Wikipedia, the free encyclopedia
A super-Poulet number is a Poulet number 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 = ... (an integer)
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 × 112 |
| 6 | 4033 = 37 × 109 |
| 7 | 4369 = 17 × 257 |
| 8 | 4681 = 31 × 151 |
| 9 | 5461 = 43 × 127 |
| 10 | 7957 = 73 × 109 |
| 11 | 8321 = 53 × 157 |
[edit] 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.
