[seqfan] A188803

[Tw Mike]

Dear seqfans,

I get a seq 17, 257, 641, 65537 recently(167772161 also is item), it
looks like A188803,
A188803 Primes that divide 2^(2^100) - 1 and 10^(10^100) - 1.
3, 17, 257, 641, 65537, 167772161, ...,

My seq is positive odd integer n that matchs this three conditions:
1. The period length of the decimal expansion of 1/n=2^x and n−1
divides by the length.
2. The sum of n=2^x.
3. The cycle length of n=2^x.
(x is some positive integer)

(using n=23 as an example to define the sum of n and the cycle length of n):
Step 1 : Get the odd part of 23+1, which is 3, 3×2^3=23+1,get s1=3
Step 2 : Get the odd part of 23+3, which is 13,13×2^1=23+3,get s2=1
Step 3 : Get the odd part of 23+13, which is 9, 9×2^2=23+13,get s3=2
Step 4 : Get the odd part of 23+9, which is 1, 1×2^5=23+9,get s4=5

Continuing this operation (with 23+1) repeats the same steps as above.
There are 4 steps in the cycle, so the cycle length of 23 is 4，and the
sum of 23 is s1+s2+s3+s4=11.
The period length of the decimal expansion of 1/23 is 22.

What's the relationship between this two seqs?

Best regards
Mike