[seqfan] Re: patterns continue for larger numbers? generalised question

[Neil Fernandez]

In message <R+qZFSA2MemdFweC at borve.org>, Neil Fernandez
<primeness at borve.org> writes

>The definition of the sequence (n,t,u) (with t,u > 1) is this
>
>a(1) = n
>a(i+1) = a(i)/t if t|a(i),
>   otherwise a(i+1)= a(i) + smallest uth power > a(i)
>
>So Ali's sequences are (n,2,2), which we know always come back to 1 or
>11 for n <= 1200000.
>
>Is there an (n,t,u) that can be shown to explode?

In answer to that question, I have found two possible candidates.

Results so far:

* for (n,2,2) and n < 1.2*10^6, all n eventually reach one of two
cycles, namely

(1,5,14,7,16,8,4,2) or
(11,27,63,127,271,560,280,140,70,35,71,152,76,38,19,44,22).

* for (n,2,3) and n < 10^5, all n eventually reach one of five cycles,
namely

(5,13,40,20,10)
(9,36,18)
(27,91,216,108,54)
(147,363,875,1875,4072,2036,1018,509,1021,2352,1176,588,294) or
(1565, 3293, 6668, 3334, 1667, 3395, 7491, 15491, 31116, 15558, 7779,
15779, 33355, 69292, 34646, 17323, 34899, 70836, 35418, 17709, 37392,
18696, 9348, 4674, 2337, 5081, 10913, 23080, 11540, 5770, 2885, 6260,
3130)

* for (n,3,2) and n < 1000, all n other than 907 and 925 eventually
reach one of three cycles, namely

(2,6)
(10, 26, 62, 126, 42, 14, 30) or
(427, 868, 1768, 3617, 7338, 2446, 4946, 9987, 3329, 6693, 2231, 4535,
9159, 3053, 6189, 2063, 4179, 1393, 2837, 5753, 11529, 3843, 1281)

In each of the cases (907,3,2) and (925,3,2) I have not been able to
discover whether the sequence reaches one of these cycles, reaches
another cycle, or explodes.


Neil

-- 
Neil Fernandez