Recursive Sequence: Highest Prime Divisors

[Leroy Quet]

If we start with two positive integers {m,n}, and let
a(1) = m,
a(2) = n,

and we let, for k >= 3,

a(k) = 
(highest prime dividing a(k-1)) + 
(highest prime dividing a(k-2)),


then we get a (possibly eventually-periodic) sequence. 

(For example: m = 2, n = 3, leads to ->

2, 3, 5, 8, 7, 9, 10, 8, 7, 9,...)

I am not at all certain, but I wonder if every {m,n} leads to a sequence 
that is eventually periodic.

Is this so??

Obviously, if the highest primes dividing a(j) and a(j-1) are those which 
are the highest primes which also divide a(k) and a(k-1), respectively 
(for some k not = j), then the sequence is periodic.

There may be something in the EIS, but the fact that {m,n} can vary makes 
a search not too easy, since I do not know which {m,n}-sequence is 
fundamental enough to be in the EIS, yet such a sequence is not trivial.

Thanks,
Leroy Quet