The Collatz Conjecture and the Hailstone Sequence

Stopping Times of the Hailstone Sequence

Click here to see a graph of the hailstone sequence stopping times.

The Collatz conjecture is named after Lothar Collatz who first proposed it in 1937. This conjecture states that the hailstone sequence for any positive integer always converges to one. The hailstone sequence is a series of numbers generated using the following simple rules. The next number in the series is calculated by dividing the current number by two if the current number is even or multiplying by three and adding one if the current number is odd. For example, the hailstone sequence for 17 is 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. The Collatz conjecture has never been proven.

This javascript-generated graphic plots the total stopping times (vertical axis scaled 1 to 300) for the hailstone sequence generated for the first 10000 positive integers (horizontal axis). The total stopping time is defined as the number of elements in the hailstone sequence before it converges to one. For example, the total stopping time for 17 is 13.

While the Collatz conjecture has never been proven, to me the real mystery lies in the patterns evident in the total stopping time plot. Why are there periodic horizontal striations? What is special about the numbers yielding large outliers? Why are there so many irregularities in the midst of so much regularity? Why are the horizontal striations staggered vertically forming interweaving bands?

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