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