[seqfan] Re: Will this pattern continue for all numbers?
Tim Peters
tim.peters at gmail.com
Fri Apr 17 22:04:03 CEST 2020
[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 ;-)
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