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

[Tim Peters]

[David Seal]
> As a terminology suggestion, perhaps "merging point" would
> be better than "meeting point"? In terms of the natural meaning
> of the words, "merging point" implies that the two sequences
> become the same from that point on, like e.g. two rivers merging,
> whereas "meeting point" only implies they're the same at that
> point, like e.g. two friends meeting.

Which is why I stuck with the original "meeting" ;-)  The idea that
the sequences become identical upon the first match was advanced in
the original post, and not questioned, but it's not true.  Indeed,
both sequences - by construction - _start_ at the same number.  They
diverge later.  The same can happen at the next time they match.

I think the smallest example starts with 95.  Doing smallest prime first, we get

95, 100, 105, 108, 111, 114, 133, 140, 147, 150, 155,  ...

Largest prime first,

95, 114, 116, 145, 150, 155, 160, 165, 168, 175. 180, ...

They first meet at 114.  Then they diverge again.  They next meet at
150.  _Then_ they're the same forever after.  It depends on whether
the parities of the numbers of iterations we've done to get to the
match are the same - i.e., on whether both sequences are or aren't
going to do the same of "largest prime" or "smallest prime" next
(which may become more complicated than just that if the meeting point
is a power of a prime).


>> On 17 April 2020 at 21:04 Tim Peters <tim.peters at gmail.com> wrote:
>>
>>
>> [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.