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The parity sequence is the same as the sequence of operations. Using this form for f ( n ) , it can be shown that the parity sequences for two numbers m and n will agree in the first k terms if and only if m and n are equivalent modulo 2 k .
English: This is a graph, generated in bottom-up fashion, of the orbits of all numbers under the Collatz map with an orbit length of 20 or less. Created with Graphviz, with the help of this Python program: # This python script generates a graph that shows 20 levels of the Collatz Conjecture.
This template's documentation is missing, inadequate, or does not accurately describe its functionality or the parameters in its code. Please help to expand and improve it . The above documentation is transcluded from Template:Collatz sequence Generator/doc .
An algorithm is fundamentally a set of rules or defined procedures that is typically designed and used to solve a specific problem or a broad set of problems.. Broadly, algorithms define process(es), sets of rules, or methodologies that are to be followed in calculations, data processing, data mining, pattern recognition, automated reasoning or other problem-solving operations.
In the original Collatz sequence, the successor of n is either n / 2 (for even n) or 3n + 1 (for odd n). The value 3n + 1 is clearly even for odd n, hence the next term after 3n + 1 is surely 3n + 1 / 2 . In the sequence computed by the tag system below we skip this intermediate step, hence the successor of n is 3n + 1 / 2 ...
In number theory, Kaprekar's routine is an iterative algorithm named after its inventor, Indian mathematician D. R. Kaprekar. [1] [2] Each iteration starts with a number, sorts the digits into descending and ascending order, and calculates the difference between the two new numbers.
In contrast, look at how useful the heuristic is: every odd number in a Collatz sequence will be succeeded by a mean of two even numbers. Earlier, I showed how to predict the number of odd numbers in the Collatz sequence of 2**177149-1 and got an answer (853681) that was within 0.12% of the actual result of 854697.