[seqfan] Re: Iterating "smallest odd prime divisor of n^2 + 1"

[Peter Munn]

Dear Frank and SeqFans,

I'd like to suggest a different way of looking at it probabilistically. 
Isn't the proportion of numbers with least odd prime factor p
approximately c_1 / (p log p) for some constant c_1?  I can see this being
the key element in the "probability" of a prime p being a term in a cycle,
leading to that probability being about c_2 / (p log p) for some c_2.

Again, I'm not so experienced as to be sure, so I ask rather than state:
does c_2 / (p log p) sum to infinity over primes p, with partial sums
convergent to c_3 log log p, or some similar formula?  If so, why not an
infinite, if sparsely distributed, number of cycles like (5 <-> 13)?

The answer doesn't seem obvious and I'd appreciate reading further comments.

Best Regards, Peter

On Sat, November 4, 2017 10:45 am, Frank Adams-Watters via SeqFan wrote:
> It was definitely my impression that it did always fall into that loop
> when I looked at the question in conjunction with my edit to the related
> sequence A031439 back in 06.  I didn't have any idea of how to prove it,
> however.
>
> Probabilistically, it seems likely: you only have to get a term congruent
> to 2 or 3 mod 5, and you immediately fall into the loop.
>
> Franklin T. Adams-Watters
>
>
> -----Original Message-----
> From: Neil Sloane <njasloane at gmail.com>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Sent: Fri, Nov 3, 2017 11:59 pm
> Subject: [seqfan] Iterating "smallest odd prime divisor of n^2 + 1"
>
> Dear Seq Fans, While I was at Hofstra Univ. the other day, Zoran Sunik
> asked, if you iterate the map "n -> smallest odd prime divisor of n^2+1",
> do you always end in the 2-cycle (5 <-> 13) ?Does anyone know?
> See A125256 for the map, and also its bisections A256970and A293958.
> If this is true, then there could be a sequence giving the number of
> iterations needed to reach the loop, or to reach any loop if there are
> others ...
>
> --Seqfan Mailing list - http://list.seqfan.eu/
>
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