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Are there sequences of numbers with identical Collatz lengths?
In fact, there are probably arbitrary long sequences of consecutive numbers with identical Collatz lengths. Here’s a heuristic argument: A number n usually takes on the order of ~ log(n) Collatz steps to reach 1. Suppose all of the numbers between 1 and n have random Collatz lengths between 1 and ~ log(n).
How did the Collatz sequence get its name?
These playful sequences, named after German mathematician Lothar Collatz (1910–1990), cause mathematicians a lot of headaches. What is most interesting about this problem is how a simple set of rules can create intricate complexity. The following iterative sequence is defined for the set of positive integers:
Which is the longest sequence of consecutive numbers?
TL;DR: between 1 and n, the longest sequence of consecutive numbers with identical Collatz lengths is on the order of log ( n) loglog ( n) numbers long. I wrote a java program which finds long consecutive sequences, here’s the longest I’ve found so far.
How many Collatz sequences can be generated in Project Euler?
Generating one million Collatz sequences and finding the longest one requires a lot more than a minute of processing time allowed for in Project Euler. The second version of the code is optimised because it stores the length of all sequences in an array.
How many operations are there in the Collatz conjecture?
There are three operations in collatz conjecture ( + 1, ∗ 3, / 2 ). The + 1 and / 2 only change the right most portion of the number, so only the ∗ 3 operator changes the left leading 1 in the number.
Which is the longest sequence of consecutive numbers in Java?
I wrote a java program which finds long consecutive sequences, here’s the longest I’ve found so far. It has 126 consecutive sequence lengths. Oddly enough, the sequence length for the number before and the number after are both 173. I’ll paste my code down below.