Hasse diagrams

This is a piece of the class of amusing mathematical diversions.

I'm working on a problem involving random directed graphs and use the concept of a Hasse diagram: it's just a graph representing a partial order in a minimal way. I stumbled across a site which draws Hasse diagrams of the relation i divides j, where i and j are positive integers. I tried it for various numbers. For example, the Hasse diagram corresponding to the divisors of 2010 is a graph with constant degree equal to 4. Whereas 2009 does not have this property. Besides the obvious significance in numerology [yes, this is a joke], there is a natural question as to what kind of numbers have the property that their Hasse diagram has constant degree.

The page above is part of what seems to be a nice undergraduate book on Algebra, titled Interactive Algebra, by A.M. Cohen, H. Cuypers and H. Sterk.