[seqfan] On periodic sequences

[Tomasz Ordowski]

Dear SeqFans!

Here is the recursive definition of my sequences:

a(1) is an od prime;
a(n+1) is the smallest prime p <> a(n) such that 2^(p-1) == 1 (mod a(n)).

Theorem: If this sequence is bounded, then it must be periodic.

Conjecture: Each such sequence is bounded.

The sequences (and their periods):

3, 5, 13, (37, 73, 19), ...
5, 13, (37, 73, 19), ...
7, 13, (37, 73, 19), ...
(11, 31), ...
13, (37, 73, 19), ...
17, 41, 61, 181, 541, 1621, 4861, 2917, 3889, 1297, 2593, 163, (487, 1459),
...
(19, 37, 73), ...
(23, 67, 199, 397, 89), ...
(29, 113), ...
(31, 11), ...
(37, 73, 19), ...
41, 61, 181, 541, 1621, 4861, 2917, 3889, 1297, 2593, 163, (487, 1459), ...
43, (29, 113), ...
47, 139, 277, 461, 1381, (5521, 16561), ...
(53, 157), ...
(59, 233), ...
61, 181, 541, 1621, 4861, 2917, 3889, 1297, 2593, 163, (487, 1459), ...
(67, 199, 397, 89, 23), ...
71, 211, 421, 2521, 6301, (4201, 1051, 701, 2801), ...
(73, 19, 37), ...
...

The odd primes a(1) that give a clean period are

11, 19, 23, 29, 31, 37, 53, 59, 67, 73, 89, 97, 101, 103, 113, ...

The period of each such sequence begins with one of these primes.

These primes are not in the OEIS. Please more terms.

What is their proportion of all odd primes?

See also https://oeis.org/A321992

Best regards,

Thomas Ordowski
___________
Amiram Eldar confirmed the periodicity of all my sequences for a(1) up to
3863.

3863, 11587, 23173, 115861, 1274461, 7646761, 38233801, 152935201,
535273201, 1338183001, 2676366001, 93672811, 200727451, 602182351,
(9634917601, 19269835201), ...