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:

• $n\to {\displaystyle \frac{n}{2}}$ if $2|n$
• $n\to {\displaystyle \frac{n}{3}}$ if $3|n$
• $n\to {\displaystyle \frac{n}{5}}$ if $5|n$
   $\dots$
• $n\to {\displaystyle \frac{n}{p-1}}$ if $(p-1)|n$
• $n\to p\times n+1$ $\text{ if none of the primes less than }p\text{ divide }n$

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:

• $n\to {\displaystyle \frac{n}{2}}$ if $n$ is even
• $n\to p\times n+1$ 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:$$3\to 1\to 30\to 15\to 5\to 1$$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