Being fascinated by the pseudo-equality problem solved in https://github.com/vincenzoaltavilla/fib-seq-vs-miles-to-kms, I asked Chat GPT to tell me other pseudo-equalities between natural numbers.

It answered me with this great statement: A number chain is created by continuously adding the square of the digits in a number to form a new number until it has been seen before.

For example,

44→32→13→10→1→1

85→89→145→42→20→4→16→37→58→89

Therefore any chain that arrives at 1 or 89 will become stuck in an endless loop. What is most amazing is that EVERY starting number will eventually arrive at 1 or 89. In this program I also checked the maximum number of steps to arrive to 1 or 89.

I noticed the motivation here: https://math.stackexchange.com/questions/1933385/repeated-sum-of-square-of-digits-always-arrives-at-1-or-89.