# 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

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/

```