This is the sequence of degrees of primitive groups which don’t synchronize a map of rank 3, equivalently graphs with clique number and chromatic number 3 having primitive automorphism groups. You could argue that the sequence should start with 3, but this is a trivial case.

The numbers must all be multiples of 3, since the colouring has all colour classes of equal size. But all numbers above are odd, and all except 21 are multiples of 9; I don’t know why this is.

How does the sequence go on? It contains 243, 441, 729, … but I don’t know if there are other terms before these.

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About Peter Cameron

I count all the things that need to be counted.

In advance of getting the complete list of endomorphisms of the 45-vertex graph from James, I will stick out my neck and say, not only does the number of vertices appear to be even, but the ranks of endomorphisms also appear to be even,

Gordon added another number to the list — 495 — and also showed that with this addition it is complete up to 729.