Finding the smallest starting number producing a Collatz sequence of length 125 which ends with a value > 4

Question

I'm supposed to do a Mathematica function that selects the smallest number not less or equal to 4 after 125 iterations. I am not good at coding, so some help/advice would be awesome.

My code so far.

f[m_, n_] := 
   f[m, n] = If[Divisible[f[m, n - 1], 2], f[m, n - 1]/2, 3*f[m, n - 1] + 1];
Table[f[m, i], {i, 1, 125}]

So, any tips on how to write code that lists the numbers and returns anything that gives back 4 after 125 iterations?

Answer 1

There are many questions about the Collatz sequence on this site, for example here. I interpret your question to ask for the smallest integer n that, after 125 iterations, has not settled into the 4-2-1 loop that is conjectured to terminate all Collatz iterations. You can run 125 iterations and return the result with the following definition.

collatz125[n_Integer] := Nest[If[EvenQ[#], #/2, 3*# + 1] &, n, 125]

Make a table of integer n and its result after 125 iterations, then select from the table those values of n which are not yet in the 4-2-1 loop.

Select[Table[{n, collatz125[n]}, {n, 1, 500}], #[[2]] > 4 &]

The smallest n is 313. After 125 iterations, the result is 5.

Answer 2

I decided to do this in the old-fashioned procedural way.

collatz[k_Integer?OddQ] := 3 k + 1
collatz[k_Integer?EvenQ] := k/2

collatzIter[k_ /; k > 2, n_: 125] :=
  Module[{i = 0, c0 = k, c1},
    While[++i < n,
      c1 = collatz[c0];
      If[c1 < 2, Break[]];
      c0 = c1];
    {i, c1}]

 Module[{m = 2, r = {0, 0}},
   While[r[[2]] <= 4, r = collatzIter[++m]];
   {m, r}]

{231, {125, 8}}

Thus 231 is smallest number the where Collatz sequence has an element greater then 4 at position 125, and that element is 8.

Let us check this answer.

<< "ExampleData/Collatz.m"
Collatz[231][[125]]

8

This of course means Collatz[231][[126]] is 4 and, therefore, fails by KennyColnago's interpretation of the OP's question, because he looks at the 126th element in the Collatz sequence.

Either answer can be easily modified to accommodate the other interpretation of what the OP is asking. The OP will have to tell us which interpretation is correct. In my case, I need only evaluate

Module[{m = 2, r = {0, 0}},
  While[r[[2]] <= 4, r = collatzIter[++m, 126]];
  {m, r}]

{313, {126, 5}}

to get an answer in agreement with KennyColnago's.