Collatz Lengths (Continued)

I was looking at the following function $f : N \to N$ defined as $f (n) = {\begin{cases} 1 & if n = 1 \\ \frac{n}{2} & if n is even \\ \frac{3 n + 1}{2} & otherwise \end{cases}$ and the recursive function $ℓ (n) = {\begin{cases} 0 & if n = 1 \\ 1 + ℓ (f (n)) & otherwise \end{cases}$ I am specifically interested in $ℓ (n) / \log_{2} (n)$ .

Yesterday, I made a mistake in typing in $f (n)$ and instead I defined $g (n) = {\begin{cases} 1 & if n = 1, 3 \\ \frac{n}{2} & if n is even \\ \frac{3 (n + 1)}{2} & otherwise \end{cases}$ and I didn’t notice the mistake until later. If you have experimented with Collatz lengths you should know that it is a rather interesting accident that the recursive length terminates. So, having found another function whose termination behavior is similar to the original $f (n)$ is nice.

This gave me an idea: what if I plotted the lengths of sequences coming from $f (n)$ and $g (n)$ against each other would I see that the convergence behaviors of the recursive lengths are similar.

Code

First I will need our functions:

(defun f (n)
   (cond ((= n 1) 1)
         ((evenp n) (/ n 2))
         (t (/ (1+ (* 3 n)) 2))))
F


(defun g (n)
   (cond ((< n 4) 1)
         ((evenp n) (/ n 2))
         (t (* (/ 3 2) (1+ n)))))
G

Then an iterator:

(defun iterate (fn n &optional (m 0))
   (if (= n 1)
      m
      (iterate fn (funcall fn n) (1+ m))))
ITERATE

Next, the data on lengths:

(let ((n (expt 2 22))
      (filename "collatz7Data.csv"))
   (with-open-file (out filename
                        :direction :output
                        :if-exists :supersede
                        :if-does-not-exist :create)
      (loop for i from 2 to n
            do (let ((x (log i 2)))
                   (format out "~d ~4,1f ~4,1f~%"
                               i
                               (/ (iterate #'f i) x)
                               (/ (iterate #'g i) x))))))
NIL

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