gb-sequences?

[Joshua Zucker]

On another mailing list, I got a question about gb-sequences,

given an initial prime,
if it's 2 mod 3, halve it and take the next prime,
else double it and take the next prime.

Seems to me like it should typically be a random walk, half the time
doubling and half the time halving.

Empirically it seems to always eventually reach the 17 -> 11 -> 7 -> 17 loop.

Counting # of steps to reach 7, and using Miller-Rabin probabilistic
prime tests, I find the following records for # of steps (anyone want
to try proving prime all the numbers involved in this, please go
ahead!):

(3 2)
(5 3)
(13 5)
(19 8)
(31 21)
(59 22)
(61 24)
(103 29)
(181 60)
(359 61)
(691 72)
(1367 73)
(1381 97)
(2749 100)
(3691 184)
(4021 216)
(5737 239)
(6481 451)
(9241 469)
(12211 1194)
(16963 3166)
(33871 3379)
(67699 3388)
(135367 3391)
(255457 4434)

But it always seems to reach 7 eventually.

There are some interesting "magnets" in that lots of numbers just
after 33871 seem to fall into the same sequence after a few
iterations.  I guess sometimes "next-prime" function will funnel
things up or down a little bit from the doubling or halving, so
sequences that start a short way apart will merge.

Does anyone think any sequences here are interesting enough to submit?
 (e.g., length of sequence starting with p, records in that sequence
...)

--Joshua Zucker