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

THE ESUCARYS MAPPING

 I've written about the Collatz or \(3x+1\) mapping and similar mappings in earlier posts:

• The \(3x+1\) Problem on November 1st 2016
• The Collatz Conjecture Revisited on March 15th 2018
• The \(17x+1\) Map on March 18th 2018
• The PrimeLatz Conjecture on August 25th 2019
• Palindromic Cyclops Numbers on August 10th 2020

I wasn't aware that the Collatz conjecture is also referred to an the Syracuse problem.

The Syracuse problem, also known as the Collatz conjecture or the \(3n+1\) conjecture or Ulam conjecture, is a very simple problem of arithmetics that is still unsolved today. It can be stated as follows:

Syracuse problem: \(n \geq 1\) being an integer, repeat the following operations

• If the number is even then divide it by two
• If the number is odd then multiply it by 3 and add 1

Conjecture: This process always reaches the number 1

Today, in investigating the number properties of my diurnal age (26251), I discovered that it is a member of OEIS A129133:

A129133

Numbers whose trajectory under the Esucarys map ends at the fixed point 247.

The Esucarys sequence derives its name from a reversal of "Syracuse", with the generating rule being that for the Syracuse (\(3x+1\) 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 26251, 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

Here is a permalink to the SageMath algorithm to generate this sequence. As an example, in reaching 247, 26251 follows this trajectory:

26251, 45787, 263731, 491197, 2953741, 4221688, 4480112, 6500422, 1120523, 751633, 94522, 16274, 7318, 9563, 9682, 1484, 247

Here is a permalink to the SageMath algorithm that will generate this trajectory. Figure 1 shows a graph of this trajectory.

Figure 1