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

[israel at math.ubc.ca]

Let S(x) = x + spf(x), L(x) = x + lpf(x), and F = S o L The question is 
whether F^m(x) = F^n(S(x)) for some m and n (where ^n denotes iteration n 
times). The answer is very likely yes, in fact it may be that for all 
integers x,y >= 2, F^m(x) = F^n(y) for some m and n. The point is 
essentially that for a "random" function F where F(n) > n but not too much 
greater, unless there is some good reason for the iterates of F on x and y 
to stay apart they are almost certain to meet eventually on probabilistic 
grounds. There is a complication in this case because you can have F(n) = 2 
n + 2 if n is prime. If something like this persisted, with the iterates 
increasing exponentially, the probability of a meeting would likely be less 
than 1.

I have verified that for every x from 2 to 6042, F^n(x) eventually hits 
60473. However, x=6043 does not. I don't know if F^n(6043) meets the 
iterates of 2 to 6042: it does not up to F^25464(6043) = 
963981399298868702558612537899.

Cheers,
Robert


On Apr 11 2020, Ali Sada via SeqFan wrote:

>Hi Everyone,
>
> 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.)
>
>Let's take 24, for example. With the first version we get:
>L: 24 + 3 = 27
>S: 27 + 3 = 30
>L: 30 + 5 = 35
>S: 35 + 5 = 40
>L: 40 + 5 = 45
>S: 45 + 3 = 48
>L: 48 + 3 = 51
>S: 51 + 3 = 54
>L: 54 + 3 = 57
>S: 57 + 3 = 60
>L: 60 + 5 = 65
>S: 65+ 5 = 70
>L: 70 + 7 = 77
>
> And with the second version we get: S: 24 + 2 = 26 L: 26 + 13 = 39 S: 39 
> + 3 = 42 L: 42 + 7 = 49 S: 49 + 7 = 56 L: 56 + 7 = 63 S: 63 + 3 = 66 L: 
> 66 + 11 = 77   The two versions meet at 77, and they move together after 
> that.   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
>
> Follow up sequences could be "the shortest path" or "the longest path", 
> etc. I would really appreciate it if you could help me define the 
> sequence properly, confirm the results, and work with me on follow up 
> sequences.
>
>Best,
>
>Ali
>
>
>
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>
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