sumtimes product dividing n

[David W. Wilson]

"M.V Subbarao" wrote:
> 
> I wonder if there are infinitely many integers n such that the sum of its
> digits times product of its digits is a divisor of n. On  example is n =
> 12. No doubt, we  can easily find many more.
> 
> M.V.Subbarao
> m.v.subbarao at ualberta.ca

Empirically, it looks as if every number of the form (10^3^n-1)/9 has the
desired property, which would make A049102 infinite.  Needs a proof, tho.