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

[franktaw at netscape.net]

Close, but it is not only factorizations into two factors that we are 
considering.  So, if n = product_{i=1}^k a_i with a_i < a_{i+1} for 1 
<= i < k and k > 1, there is an arrow from n to n + sum_{i=1}^k a_i.

This more restrictive definition is of some interest, though.  To 
extend Hagen's remarks, note that if n is even, everything pointed to 
by n is divisible by a smaller power of two than n; so any path will 
fairly quickly come to only odd numbers.  I think there still are 
infinite paths, but it isn't as clear as for the original problem.

Franklin T. Adams-Watters

-----Original Message-----
From: Hagen von EItzen <math at von-eitzen.de>

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 longest 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.