COLLATZ
A Simple Package for the Collatz Sequence

COLLATZ is a Mathematica package which evaluates the Collatz sequence for a given starting point.

The most interesting thing about COLLATZ is not the Collatz sequence, but rather the way that it demonstrates how a user can define new functions for Mathematica, to be stored in a package file, and accessed later by, for example, a notebook.

The Collatz sequence itself, sometimes called the "3N+1" sequence, is defined as follows. Choose a starting value n, a positive integer. Now compute the next value of n:

• If the current value of n is 1, stop. The sequence has terminated.
• If n is even, the next element is n/2;
• If n is odd, the next element is 3*n+1.

However, we are much more interested in the steps necessary to make it possible for a user to be able to start up Mathematica, and type

        Collatz[17]
      

Package Installation:

A package is a group of Mathematica function definitions. Packages have a specific format; for instance, they are delimited with BeginPackage[] and EndPackage[] statements. Moreover, (and I forget this every time!) the cells in a package must be explicitly designated as "Initialization Cells" or nothing in the package will actually be usable.

In order for a package to be accessible, it must be placed in a specific directory where Mathematica can find it. Generally, this is the Applications subdirectory of the user base directory, which can be determined by:

        $UserBaseDirectory
      

Once the new package has been placed, you should probably run Mathematica and direct it to rebuild the Help database. This will serve to update Mathematica's internal information so that it can explain how the functions in the package are called.

Package Usage:

To use the Collatz function, the user must first load the package, with the command

        << Collatz.m
      

        Collatz[99]
      

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

COLLATZ is available in a JAVA version and a MATHEMATICA version and a MATLAB version and a PERL version.

Related Data and Programs:

POLPAK is a FORTRAN90 library which includes the routine collatz_count for counting the length of a Collatz sequence.

SEQUENCE_STREAK is a MATLAB program which can be used to make a "streak plot" of a Collatz sequence for a particular starting value, or for the length of the Collatz sequence for a number of starting values.

Reference:

1. Roman Maeder,
   Programming in Mathematica,
   Second Edition, Addison Wesley, 1991.
2. Eric Weisstein,
   "The Collatz Problem",
   CRC Concise Encyclopedia of Mathematics,
   CRC 1998.
3. Stephen Wolfram,
   The Mathematica Book,
   Fourth Edition,
   Wolfram Media / Cambridge University Press, 1999.

Source Code:

• Collatz.m, the text of the Collatz package;

Examples and Tests:

• CollatzTest.nb, a Mathematica notebook that uses Collatz;

List of Routines:

• Collatz[n] gives a list of the iterates in the 3n+1 problem, starting from n. The conjecture is that this sequence always terminates.
• StoppingTime[n] finds the total stopping time for the integer n. This is the length of the Collatz sequence up to the time it reaches the value 1.
• FindMaxima[nmax] reports successive maxima for the total stopping time, starting the search with the integer nmax, and running FOREVER.

You can go up one level to the Mathematica packages and notebooks.