I was playing around with collatz sequence stuff recently, and I made a plot of seed values vs the maximum value the seed's collatz sequence will reach. Highlighting primes in a logarithmic graph gives an interesting result
In this image, we evaluate the collatz sequences of all numbers ("seed values") from $1$ to $10^5$, and find the maximum number within each sequence. We plot each seed value on the $x$ axis with its corresponding maximum number in its sequence on the $y$ axis.
Then, highlight all points corresponding to a prime seed value red, and a composite blue.
As seen in the graph, the prime numbers seem to have a consistently larger "lower bound" for their maximum values than composites.
What causes this behavior?
My initial thought was that for composites highly divisible by 2 and have numbers in its sequence that are highly divisible by 2, the maximum value is itself, hence making up the very bottom of the graph. Meanwhile, a prime is always odd so the smallest possible number in its sequence is itself scaled up by $\frac32$.
Is this the correct explanation, or is there something else I am missing?
thanks!
Best Answer
Thanks to @user140242 and @Peter for the insight.
Plotting a the same aforementioned graph but with odd numbers only shows that my explanation was pretty much correct.
The lowest possible maximum for each odd number's collatz sequence is at the very least $\frac32$ times itself. This makes up the bottom boundary for the primes as well.
Meanwhile for even composites, the lowest possible maximum could be itself, or arise later in the sequence such that it's greater than itself but still less than $\frac32$ times itself &c, making up the various other lines seen below the bottom prime boundary (shown below).
Below are some graphs, and you can see that the bottom "max number" for each odd number seed's collatz sequence is the same for composites and primes.
Primes highlighted and plotted on top
Primes just highlighted