The RATS Sequence Revisited

It was on Saturday, 26th September 2020, that I first wrote about the RATS sequence, the acronym standing for Reverse --> Add --> Then Sort. On that occasion I was looking at the number 26109 that is a member of OEIS A114613:

A114613

Starting numbers for which the RATS sequence has eventual period 3.

Yesterday the number associated with my diurnal age, 27099, is the next member of that sequence. The sequence runs (permalink):

20169, 20709, 21159, 22149, 23139, 24129, 25119, 26109, 27099, 28089, 29079, 30159, 30168, 30708, 30789, 31149, 31158, 31779, 32139, 32148, 32769, 33129, 33138, 33759, 34119, 34128, 34749, 35109, 35118, 35739, 36108, 36729, 37098, 37719, 38088, 38709, 39078, 39699, ...

As can be seen, the members of this sequence are few and far between. Up to 40,000, there are only 38 terms. In the case of 27099 the progression is:

27099, 111267, 337788, 1122255, 4446666, 1111113, 2222244, 4446666

There are sequences with different periods but for most numbers the progression is infinite. See Table 1:

Table 1

The original post contains more information that I won't repeat here. Using my diurnal age today (27100) as an example, we see the divergent period-2 pattern emerging, with the red showing the progression of this period-2 pattern (permalink).

27100
22277
49999
134999
113344
555566
1111222
2333333
5566666
11222333
44444455
88889999
178888888
156777779
113344555
666677888
1444555555
167999
116677
238889
1122277
4448888
12333337
55666688
133333444
556667777
1233334444
5566667777

12333334444

55666667777

123333334444

556666667777

1233333334444

5566666667777

12333333334444

55666666667777

123333333334444

556666666667777

1233333333334444

5566666666667777

It won't be until I turn 28089 days old that I'll meet this sequence again. The Wolfram Demonstration Project also features the RATS sequence and displays the progression of terms up till the start of the period-2 pattern. See Table 2.

Table 2