collatz

collatz, a Python code which computes the Collatz sequence.

The rules for generation of the Collatz sequence are recursive. If T is the current entry of the sequence, (T is assumed to be a positive integer), then the next entry, U is determined as follows:

1. if T is 1, terminate the sequence;
2. else if T is even, U = T/2.
3. else (if T is odd and not 1), U = 3*T+1;

Although the Collatz sequence seems to be finite for every starting point, this has not been proved. Over the range of starting values that have been examined, a great irregularity has been observed in the number of entries in the corresponding sequence.

The Collatz sequence is also known as the "hailstone" sequence or the "3n+1" sequence.

Licensing:

The information on this web page is distributed under the MIT license.

Languages:

collatz is available in a MATLAB version and an Octave version and a Python version..

Related Data and Programs:

collatz_dict, a Python code which demonstrates how the Python dict variable type can be used to efficiently record data about the Collatz iteration.

collatz_recursive, a Python code which demonstrates recursive programming by considering the simple Collatz 3n+1 problem.

polpak, a Python code which evaluates a variety of mathematical functions, polynomials, and sequences, including Bell, Benford, Bernoulli, Bernstein, Cardan, Catalan, Charlier, Chebyshev, Collatz, Delannoy, Euler, Fibonacci, Gegenbauer, Gudermannian, Hermite, Hofstadter, Jacobi, Krawtchouk, Laguerre, Lambert, Legendre, Lerch, Meixner, Mertens, Moebius, Motzkin, Phi, Sigma, Stirling, Tau, Tribonacci, Zernike.

Reference:

1. Martin Gardner,
   "Slither, 3X+1 and Other Curious Questions",
   Wheels, Life, and Other Mathematical Amusements,
   WH Freeman, 1983.
2. Brian Hayes,
   "On the ups and downs of hailstone numbers",
   Scientific American,
   January 1984,
   Volume 250, Number 1, pages 10-16.
3. Patrick Honner,
   "The Simple Math Problem We Still Can't Solve",
   Quanta,
   22 September 2022.
4. Eric Weisstein,
   "The Collatz Problem",
   CRC Concise Encyclopedia of Mathematics,
   CRC Press, 2002,
   Second edition,
   ISBN: 1584883472,
   LC: QA5.W45.

Source Code:

• collatz.py, calls all the tests.
• collatz.sh, runs all the tests.
• collatz.txt, the output file.

• collatz_27.txt, the points in the Collatz sequence for seed 27.
• collatz_count.txt, the Collatz count for seeds 1 to 100.
• collatz_sequence_27.png, a plot of the Collatz sequence for seed 27.
• collatz_max.png, a plot of the Collatz max values for seeds 1 to 1000.
• collatz_count.png, a plot of the Collatz coount values for seeds 1 to 50.