Sum-product numbers

[Eric W. Weisstein]

Does anyone have a proof on the (non)existence of any more sum-product
numbers, i.e., numbers n s.t. n = (product of digits of n) x (sum of
digits of n); http://www.treasure-troves.com/math/Sum-ProductNumber.html.  
Other than 1, 135, 144, there are no others < 10^9.  The sequence is known
to be finite
(http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=038369)  
and it seems likely that clever arguments based on divisibility might be
able to establish if any more such numbers actually exist.

Cheers,
-Eric

P.S. Does anyone have a reference where these first appeared?

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* Eric W. Weisstein                                                *
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