[seqfan] The ord-sequences

[Tomasz Ordowski]

Dear SeqFans!

Consider the ord-sequences defined by the following recursion:
a(n+1) is the smallest k > a(n) such that 2^{k-a(n)} == 1 (mod a(n)).
For example: 3, 5, 9, 15, 19, 37, 73, and 82. Ended with an even number.
Conjecture: For any initial term a(1), ord-sequence is finite (an even
term).
Note that a(n+1) = a(n) + ord_{a(n)}(2). Hence the name: ord-sequences.
Theorem: a(n+1) is even if and only if ord_{a(n)}(2) is odd. See A036259.
Question: Will each ord-sequence hit such a penultimate number?
I'm asking for new length records of my sequences (see below).

Best regards,

Thomas Ordowski
_______________
The length records:
a(1), length, last term;
1, 2, 2
3, 8, 82
11, 18, 2246
39, 33, 13192
179, 54, 811840
197, 83, 140596630
2481, 115, 1239299494
6299, 191, 263364616132
36365, 269, 240155184043288
231393, 312, 648274634463250
249971, 358, 124677735534497570
2656943, 394, ??????????????????
5006999, 439, 15748956693023917552
5745479, 477, 560480869962021796634
58182659, 523, 1051892735063463075706
[Amiram Eldar]