A Special 29x + 1 Map

 In blog posts titled:

• The 17x+1 Map on the 18th March 2018
• The 13x+1 and 11x+ 1 Maps on the 19th of March of 2018
• The 17x+1 Map Revisited on the 19th May 2025

I've looked at Collatz-style sequences of the form \(px+1\) that map a number \(n\) to the following:

• \(n \rightarrow \dfrac{n}{2}\) if \(2 \, | \, n\)
• \(n \rightarrow \dfrac{n}{3}\) if \(3 \, | \, n\)
• \(n \rightarrow \dfrac{n}{5}\) if \(5 \, | \, n\)
   \(\dots\)
• \(n \rightarrow \dfrac{n}{p-1}\) if \((p-1) \, | \, n\)
• \(n \rightarrow p \times n+1\) \( \text{ if none of the primes less than }p \text{ divide } n\)

Other mappings are of course possible and the following is perhaps the simplest of all. For a given prime \(p\) when applied to a number \(n\), the rule is:

• \(n \rightarrow \dfrac{n}{2}\) if \(n\) is even
• \(n \rightarrow p \times n + 1\) if \(n\) is odd

For \(p=3\), this is the Collatz trajectory of the number. Let's apply this sort of modified \(px+1\) mapping in the case of \(n=3\). The sequence formed by the trajectory up to one million is then (permalink):

3, 88, 44, 22, 11, 320, 160, 80, 40, 20, 10, 5, 146, 73, 2118, 1059, 30712, 15356, 7678, 3839, 111332, 55666, 27833, 807158, 403579, 11703792, 5851896, 2925948, 1462974, 731487 (OEIS A037112)

The trajectory is shown in Figure 1 using a logarithmic scale for the \(y\) axis:

Figure 1: permalink

By contrast if the first mentioned rule above were applied then the trajectory would be short-lived indeed:$$3 \rightarrow  1 \rightarrow  30 \rightarrow  15 \rightarrow 5 \rightarrow  1$$It can be seen that the trajectory very quickly enters a loop. 

Letting \(p=101\) and applying the mapping to \(n=3\) once again, we get the following sequence up to one million formed by the trajectory (permalink):

3, 304, 152, 76, 38, 19, 1920, 960, 480, 240, 120, 60, 30, 15, 1516, 758, 379, 38280, 19140, 9570, 4785, 483286, 241643

The trajectory is shown in Figure 2 using a logarithmic scale for the \(y\) axis:

Figure 2: permalink