[seqfan] Re: Add to an integer its distinct factors and loop

[Hagen von EItzen]

>
> Hello Joshua,
>
> I only jump from one integer to another one in Richard's list there :
> http://www.strw.leidenuniv.nl/~mathar/progs/A045778.txt <http://www.strw.leidenuniv.nl/%7Emathar/progs/A045778.txt> 
>
> 20-->4*5 _is_ in the table :
>
> >/ 12-->2*6--->20-->4*5--->29-->END
> /
> [my former "25-->5*5" line was wrong -- as there is a zero (0) in
>  Richard's list after 25]
>   

Ah, now I see:
We are considering the directed graph with the natural numbers as vertices
and an edge from n to m iff there are natural numbers a,b such that
1<a<b and n=a*b and m=n+a+b.
The question then is: Are there infinite paths in this graph?
(And related: How long is the llongest path starting at n? Etc.)
As already noted, primes and squares of primes are obvious end points.
Other trivial observations:
If n is not a multiple of 4, it points only to odd numbers;
If n is a power of two, it points only to even numbers;
all other numbers (i.e. multiple of 4 but not power of 2) point to 
numbers of both parities.

Hagen