Biquanimous numbers

[David W. Wilson]

I spend a couple hours on the road every day, and I have two games I
like to play with numbers on license plates.  One is factoring them
(and you thought cell phones were a menace).  The other is trying to
group the digits into two pieces with the same sum.

So, let's call a number biquanimous if its digits can be split into
two sets with the same sum.

There are some weak general laws of biquanimity:

  - Adding or removing 0 digits from a number does not affect its
    biquanimity.

  - A biquanimous number must have an even digit sum.

It is almost obvious that a base-2 number is biquanimous iff it has
an even number of 1's.

With a little work, we see that a base-3 number is biquanimous iff
either (a) it has an even number of 2's and a even number of 1's,
or (b) it has an odd number of 2's and a postive even number of 1's.

In these two bases, the biquanimous numbers form a regular language.
Another way to say this is that the base-2 and base-3 biquanimous
numbers are, respectively, 2-regular and 3-regular sets.  My gut
feeling is that this generalizes to all finite bases, though I don't
know how to start a proof.  If, however, a finite automaton could
be constructed accepting the base-10 biquanimous numbers, then they
could be very quickly counted up to any specified limit.