planar numbers?
Simon Colton
simonco at dai.ed.ac.uk
Tue Jul 25 16:14:23 CEST 2000
Dear Sequence Fans,
Has anyone heard of the following construction:
Take a number and write down its divisors, make these
the nodes of a graph. Then join any two divisors by an edge
if one divides the other. For example, 16=2^4 produces graph
K5, because it has 5 divisors, and they all divide each
other.
The question is then: which numbers produce planar
divisor graphs? Call these numbers planar numbers.
As K5 is non-planar, 16 is not planar. It turns out
that only the number 1 and integers with prime
factorisations: p, p^2, p^3, pq, p^2q are planar.
These form a sequence (not in the encyclopedia):
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,18,19,20,
21,22,23,25,26,27,28,29,31,33,34,35,37,38,39,41,....
and the non-planar numbers are:
16,24,30,32,36,40,42,48,54,56,60,64,66,70,72,78,
80,81,84,88,90,96,100,...
(also not in the encyclopedia).
I have a maple worksheet discussing this construction
and four or five others here:
http://www.dai.ed.ac.uk/~simonco/papers/planar.ms
available as a postscript file here:
http://www.dai.ed.ac.uk/~simonco/papers/planar.ps
Any comments very welcome (I'm thinking about turning
it into a J.I.S paper, although it's not very hi-tech).
All the best,
Simon Colton.
---------------------------------
http://www.dai.ed.ac.uk/~simonco/
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