Well I am quite sure it's known (I mean number theory exists thousands of years), warning beforehand, it may look like numerology, but I try not to go to mysticism.
So I was in a bus, and from boredom I started just adding numbers in the next way: $$1+1=2$$ $$2+2=4$$ $$4+4=8....$$
etc up to $32,768$ (it was quite boring, I can tell... :-)), I didn't have calculator.
and notice that if I keep adding the digits until I get a number from 1 to 10, I get that for example for $8+8=16$, $1+6=7$, now seven steps after this, at $512+512=1024$, which $1+2+4=7$, and again after $7$ steps $32768+32768=65536$, and adding $6+5+5+3+6=10+12+3=25$, $2+5=7$. So this led me to cojecture that this repetition may occur endlessly.
Now ofcourse I can programme some code that will check for large numbers, but I am tired, long day. So if this indeed the case (which could be disproved, but even then I would wonder when this repetition stops) then why?
As I said, I am tired, it may make no sense, and I might have done mistakes in my calculations, and it may be trivial.
Either way, if you have some answer, I would like to hear it.