Sequence $ 17, 257, 641, 65537, … $ It consists of odd positive integers that match these three conditions:

- The period duration of the decimal expansion of $ 1 / n = 2 ^ x $ and divide $ n-1 $.
- The sum of $ n = 2 ^ x $.
- The cycle duration of $ n = 2 ^ x $.

($ x $ it's a positive integer)

(using $ n = $ 23 as an example to define the sum of $ n $ and the cycle duration of $ n $):

Step 1: Get the strange part of $ 23 + ~~ $ 1, which $ ~~ $ 3,$ ~~ 3 times2 ^ 3 = 23 + ~~ 1 $,obtain $ s_1 = $ 3

Step 2: Get the strange part of $ 23 + ~~ $ 3, which $ 13 $,$ 13 times2 ^ 1 = 23 + ~~ 3 $,obtain $ s_2 = 1 $

Step 3: Get the strange part of $ 23 + $ 13, which $ ~~ $ 9,$ ~~ 9 times2 ^ 2 = 23 + 13 $,obtain $ s_3 = 2 $

Step 4: Get the strange part of $ 23 + ~~ $ 9, which $ ~~ $ 1,$ ~~ 1 times2 ^ 5 = 23 + ~~ 9 $,obtain $ s_4 = 5 $

Continuing this operation (with $ 23 + $ 1) Repeat the same steps as above.

exist $ 4 $ steps in the cycle, so the cycle duration of $ 23 $ is $ 4 $， And the sum of $ 23 $ is $ s_1 + s_2 + s_3 + s_4 = $ 11.

It seems that all known elements of the sequence are Fermat factors, how is that?