Recursive Sequence: Highest Prime Divisors
Leroy Quet
qqquet at mindspring.com
Tue Sep 16 04:41:55 CEST 2003
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
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