[seqfan] Re: Will this pattern continue for all numbers?

[Tim Peters]

[Ali Sada via SeqFan <seqfan at list.seqfan.eu>]
> We start with n and use the map k-->k+p, where p alternates between
> the “largest prime factor” and the “smallest prime factor”.
>
> We have two versions here: a)    largest, smallest, largest, etc.; and
> b)    smallest, largest, smallest, etc.
>
> What is interesting is that it seems, at least for the small group of
> numbers I checked, that no matter what version we use, we will
> reach a meeting point. (I can't prove that.)

"Meeting point" needs some elaboration.  When k is a power of a prime,
(a) and (b) meet on their first iterations.  But the results later
seem to skip over all initial "meetings", waiting until (a) and (b)
diverge before a later "meeting" counts.  For example, 2 maps to 4
either way, and then 4 maps to 6 either way, but neither of those seem
to count.

> ...

> The sequence associated with this algorithm is the “meeting point” for each
>  number (starting from 2):
>  12, 12, 12, 15, 77, 30, 15, 21, 15, 77, 21, 77, 21, 77, 30, 77, 30, 77, 77,  30, 77, 91, 77, 51, 77, 77, 77, 105

And later Luca Petrone said he got 12 for k=6 instead of 77, and you
confirmed that.

Hans Haverman also got different results for 14, 19, 25, and 28.  He
didn't say what he got instead, and I didn't see a response to him.
So here's what I got ;-)

12, 12, 12, 15, 12, 30, 15, 21, 15, 77, 21, 77, 30, 77, 30, 77, 30,
57, 77, 30, 77, 91, 77, 77, 77, 77, 35, 105

That disagrees at the same positions Hans found disagreements.

Dr. Israel found it easy through k=6042, but 6043 looks tough.  I let
iterates get about 1000 times larger than where he stopped, but still
didn't find a meeting point.  The time for factoring is a real drag
then (although I'm using nothing fancier than Pollard rho for that).

So is 6043 uniquely hard - or common?  Alas, it's not unique.  Trying
larger starting points, they seem to fall in two categories:  "goes
fast" and "no end in sight"

Here are the smallest I found before giving up:

6043, 6551, 6791, 7057, 7177, 7213, 7229, 7411, 7499, 7529, 7541, 7547

Those are all primes.  Is it limited to primes?  I think that's a
natural question, since all iterates are composite (k+p is necessarily
divisible by p. since p divides k by construction).

So I tried changing the definition of "meeting point" to NOT overlook
initial meetings (so that powers of primes found a meeting point at
once), and the smallest hard case then was 11589 (3 * 3863), and next
after that 11733 (3 * 3911).

So:  no insights or striking patterns from me!  Those also serve who
only thrash and fail ;-)