Smith Numbers and Rhonda Numbers

Albert Wilansky once noticed that the phone number of his brother-in-
law, 4937775, had the property that the sum of the digits equals the
sum of the digits of the prime divisors.  For another example, the 
integer 6036 = (2)(2)(3)(503) has this same property, because 
6+0+3+6 equals 2+2+3+5+0+3.  Wilansky called these "Smith numbers",
after his brother-in-law, H. Smith.  We might ask whether such numbers
are a fit subject for serious work, raising again the old question of
what constitutes "serious" mathematics, as opposed to "recreational"
mathematics.

Whenever this question is discussed, someone dutifully explains that
anything having to do with the decimal digits of a number is not 
considered serious math, because those digits are just a consequence
of the arbitrarily chosen base 10, and don't represent anything 
significant.  We might say it is analagous to a physicist studying 
the decimal digits of the mass of a proton in kilograms.  G. H. Hardy
frequently mentioned the study of properties of decimal digits as an 
example of what he considered to be non-serious math.  However, many
people go on to suggest that the criterion of seriousness is USEFULNESS,
whereas Hardy certainly would not have agreed with that, considering 
his well known views on the "usefulness" of his work - he claimed 
with pride that he had never done anything useful in his life, 
although he clearly regarded his mathematics as serious.

Apart from strict utilitarians, I think most people would say that
"seriousness" of math is largely a matter of esthetic judgement, so
there's not much point in arguing about it (like what's serious art?)  
On the other hand, I wonder if we are too quick to dismiss subjects 
like the properties of decimal representations, both from a utilitarian 
and an esthetic standpoint.  

The mass of a proton in kilograms is surely arbitrary, because it 
is simply the ratio of the proton's mass to a reference mass (the 
kilogram) that has no particular physical significance (aside from 
being a convenient order of magnitude for many of the objects with 
which we interact).  However, in the study of numbers themselves, it 
isn't clear to me why the base number is any less "significant" than 
the number being represented.  For any choice of the base B, each 
integer N has a unique representation as a sum of powers of B times 
positive coefficients less than B.  Doesn't this express a real and
absolute property of the pure numbers N and B?  Doesn't this provide 
a rich structure for study?  Why should this type of structure 
necessarily be considered non-serious?

Two possible answers are (1) decimal representations are well known 
to non-mathematicians, and mathematicians prefer to work on things 
that are unfamiliar to non-mathematicians, and (2) operations expressed 
in terms of base-B representations are so intractable that it's hard 
to prove theorems about them.  

To illustrate the 2nd point, consider whether every reverse-sum 
sequence in base 10 leads to a palindrome number.  Nobody knows.  
Adding a number such as 3247 to its reversal 7423 is very simple, 
but it's not nearly so simple to express the function N+rev(N) in 
a tractable form that could be iterated and used to prove theorems.  
We have
            N  =  7  +  4 B  +  2 B^2  +  3 B^3

which can also be written in the form

            N  =  3 (r1 + B)(r2 + B)(r3 + B)

where r1,r2,r3 are the roots of the monic polynomial in B for N/3.  
Then rev(N) can be expressed as a polynomial with the inverse roots 
1/r1, 1/r2, and 1/r3. Specifying each representation in terms of its 
roots instead of its coefficients allows us to make some progress, 
but the reconstruction of the coefficients with the effects of 
borrowing and carrying is still extremely difficult.  The fact is 
that nobody knows a very powerful way of manipulating this kind of 
mathematical structure, even though we use it every day to balance 
our checkbooks.  So we dismiss it as "not serious".

Well, anyway, the brother-in-law's phone number raises another 
question.  How unusual is it to find a number with some peculiar 
property?  Is it possible to find an interesting property for ANY 
number?  This reminded me of the Rhonda Numbers.  The address of a 
former acquaintance of mine was 25662 W. Something.  The prime 
factorization of 25662 is (2)(3)(7)(13)(47), so the sum of the prime 
factors is 72.  The product of the decimal digits is 720, which is B 
times 72.  Thus, a Rhonda Number in the base B is a number such that 
the product of the digits in base B equals B times the sum of the
prime factors.  (Of course, B must be composite for there to be any 
Rhonda Numbers to the base B.)  In the base 10 there are 38 Rhonda 
Numbers less than 200,000.  The number 1000 is a RN in base 16, and 
the number 3000 is an RN in the base 18.  Ok, so my question is:  
Can a number N be a Rhonda Number relative to more than one base?
(For an answer to this question, see Infinitely Many Rhondas.)