Base tree

Odd Numbers And Reachable Branches

This tree lists all odd numbers, shows their forward Collatz even value with 3n + 1, and marks the odd multiples of 3 as the class that cannot be reached from another odd number in the inverse odd-to-odd tree.

Proof Use

Since 3n + 1 is never divisible by 3, dividing it by powers of 2 cannot produce an odd multiple of 3. That means only two out of the three odd residue classes are reachable from earlier odd values; the 3 mod 6 odd numbers can appear only as starting points.

The reachable odd lanes are exactly 6x - 1 and 6x + 1. These are the same lanes that contain every prime greater than 3, so the usable odd numbers are prime candidates, while still including composite values such as 25 and 35.

odd parent m = (n * 2^k - 1) / 3, valid only when this is an odd integer

Odd rows Doubling columns

0 mod 6 1 mod 6 2 mod 6 3 mod 6 4 mod 6 5 mod 6