Recursive Sequence: Highest Prime Divisors

[Neil Fernandez]

In message <E19z5oZ-0005ss-00 at hall.mail.mindspring.net>, Leroy Quet
<qqquet at mindspring.com> writes

>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.

I conjecture that if m and n are co-prime, all such sequences eventually
start repeating 10, 8, 7, 9, 10, 8, 7, 9, ...

Mathematica:
t = 2; u = 3; Do[
  Print[First[Last[FactorInteger[t]]] + First[Last[FactorInteger[u]]]]; 
  v = u;
  u = First[Last[FactorInteger[t]]] + First[Last[FactorInteger[u]]]; 
  t = v, {20}]

Neil

-- 
Neil Fernandez