Step by Step solution about How to resolve the algorithm Juggler sequence step by step in the Mathematica / Wolfram Language programming language

This code, written in the Wolfram programming language, investigates the behavior of a sequence of integers generated using a specific rule. The sequence starts with an input integer, n, and the subsequent terms are calculated using the next function. The stats function then analyzes the sequence and provides various statistics.

next Function:

• This function takes an integer n as input.
• If n is even, it calculates the floor (rounded down value) of the square root of n.
• If n is odd, it calculates the floor of n^(3/2).
• The output of this function is the next term in the sequence.

stats Function:

• This function takes an integer n as input and generates a sequence of integers using the next function.
• It stops the generation when the sequence reaches 1.
• The function then performs the following:
  • It stores the generated sequence in the variable data.
  • It calculates the maximum value in the sequence and its position using Ordering.
  • It returns a list of statistics: n, the length of the sequence, the maximum value, and the position of the maximum value.

Table Generation:

• The code generates a table of statistics for a range of input integers (n) from 20 to 39.
• For each n, it calls the stats function and stores the results in a list.
• The list of statistics is then used to create a table that is formatted using TableForm.
• The table headings are set to "n", "length", "max", and "max pos".

Output: The output of the code is a table that shows the statistics for each input integer. This table can be used to analyze the behavior of the sequence generated by the next function.

next[n_Integer] := If[EvenQ@n, Floor[Sqrt[n]], Floor[n^(3/2)]]

stats[n_Integer] :=
 
 Block[{data = Most@NestWhileList[next, n, # > 1 &], mx},
  mx = First@Ordering[data, -1];
  {n, Length[data], data[[mx]], mx - 1}]
  
{TableForm[Table[stats@n, {n, 20, 39}], 
 TableHeadings -> {None, {"n", "length", "max", "max pos"}}]