The Esucarys Mapping Revisited

It was on the 15th February 2021 that I made my first post about the Esucarys Mapping which is related the Collatz or 3\(x\)+1 mapping but with an extra twist. Let's revisit what I wrote back then.

The Esucarys sequence derives its name from a reversal of "Syracuse", with the generating rule being that for the Syracuse (3\(x\)+1 or Collatz) sequence followed by a reversal. 247 is the only known fixed point of the Esucarys sequence. Very few numbers map to 247.

The members of this sequence, up to 40000, are:

247, 1247, 1484, 2473, 4859, 5087, 5738, 7318, 7484, 9563, 9682, 9694, 9938, 11247, 12189, 12473, 14840, 14842, 15209, 15610, 16274, 16563, 16750, 16798, 17609, 19168, 20019, 21885, 24733, 26251, 27123, 27125, 29156, 30076, 30524, 32614

Back when I made that post my diurnal age was 26251 and it was only today that my diurnal age reached the next term, 27125, in this sequence (OEIS A129133). This latter number requires only five steps to reach 247. The steps are:

27123, 7318, 9563, 9682, 1484, 247

The trajectory is shown in Figure 1.

Figure 1

The progression reached thus:

• 27123 --> 81370 (multiply by 3 & add 1 since number is odd)
• 81370 --> 7318 (reverse number)
• 7318 --> 3659 (divide by two since number is even)
• 3659 --> 9563 (reverse number)
• 9563 --> 28690 (multiply by 3 and add 1 since number is odd)
• 28690 --> 9682 (reverse number)
• 9682 --> 4841 (divide by 2 since number is even)
• 4841 --> 1484 (reverse number)
• 1484 --> 742 (divide by 2 since number is even)
• 742 --> 247 (reverse number)

Since 247 --> 742 --> 247 we are stuck. Note that certain numbers will produce infinite loops but they don't centre on a fixed point. For example consider the numbers 3 and 13:

3 --> 10 --> 1 --> 4 --> 2 --> 1 --> 4

13 --> 40 --> 4 --> 2 --> 1 --> 4

What's interesting about 27123 is that the next odd number, 27125, is also a member of the sequence. The progression for this number also involves five steps and is:

27125, 67318, 95633, 9682, 1484, 247

The maximum values reached by the trajectory of both these numbers is tiny compared to the previous singleton (26251, see earlier post). Pairings like 27123 and 27125 are relatively rare. Up to one hundred thousand, the only ones are:

• 14840 and 14842
• 27123 and 27125
• 74840 and 74842
• 82823 and 82825

 Figure 2 shows a plot of the sequence members up to one hundred thousand.

Figure 2