The only possible adjacent positive integer pairs with a 
product less than 4 are "12", "21", "13", and "31".

So we start with 6!=720 arrangements of 1,2,3,4,5,6.

Then we subtract the 5!=120 arrangements of (12),3,4,5,6 and
the 5!=120 arrangements of (21),3,4,5,6.

Then we subtract the 5!=120 arrangements of (13),2,4,5,6 and
the 5!=120 arrangements of (31),2,4,5,6.

But we have subtracted out those containing "213" twice 
and those containing "312" twice.  So we need to add back in
the 4!=24 arrangements of (213),4,5,6, and the 4!=24 
arrangements of (312),4,5,6, once each.

So the answer is

6! - 4x5! + 2x4! = 288.

Edwin