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Which operation is performed, 3n + 1 / 2 or n / 2 , depends on the parity. 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. This implies that ...
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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 ...
Lothar Collatz (German:; July 6, 1910 – September 26, 1990) was a German mathematician, born in Arnsberg, Westphalia. The "3x + 1" problem is also known as the Collatz conjecture, named after him and still unsolved. The Collatz–Wielandt formula for the Perron–Frobenius eigenvalue of a positive square matrix was also named after him.
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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.