19.8.16

Collatz Primes

Although I found a counter example to the last claim from my previous Collatz adventure, which means that it is false in general, I decided to look for which products of primes it is true. It turns out that from all products of primes for primes from 5 to 37 this claim is true for almost all products.

In other words, almost all of these products of primes are reduced to either one of the primes of which they are composed, or is a multiple thereof when ((n+1)/2 is subjected to the algorithm behind the Collatz Conjecture. 

Obviously I found this fascinating but not particularly clean or elegant. So I decided to look what happens to individual odd primes p when (p+1)/2 is subjected to the same algorithm.

Definition: A Collatz prime (or a selfie prime) is an odd prime p such that p+1 reaches p when put through the algorithm behind the Collatz conjecture.

The first few such primes are:

5, 13, 17, 53, 61, 107, 251


Note: This sequence is currently not in the encyclopedia of integer sequences.

Observation:  It appears that not all products of Collatz primes are reducible to the primes of which they are composed and yet many products of primes that are not Collatz primes can still be reduced to the primes of which they are composed

Observation 2: The distribution of these primes is strange, and I haven't yet been able to find another Collatz prime after 251 

Below is a simple algorithm to check if an odd prime is a Collatz prime