planar numbers?

Simon Colton simonco at
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

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):


and the non-planar numbers are:


(also not in the encyclopedia).

I have a maple worksheet discussing this construction
and four or five others here:

available as a postscript file here:

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.

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