> Of course, this really only works well for numbers with prime factors drawn from the set {2, 3, 5, 7}.

So express the number as a sum of numbers with factors drawn from that set, and draw a several diagrams, one for each summand. A quick bit of Python confirms that every number under 14925 can be expressed as the sum of no more than three such numbers in the most obvious way possible (ie using a greedy algorithm).

For example, you might write 9999 as 9800 + 196 + 3, all of which have nice factorization diagrams.