Sum-Product numbers

[Giovanni Resta]

Eric W. Weisstein wrote:

> OEIS
> 
> http://www.research.att.com/~njas/sequences/A114457
> 
> gives the smallest k such that abs(S(k)P(k)-k) equals n, where S(k) is 
> the sum and P(k) is the product of decimal digits of k.
> 
> a(n) is small for 0<=n<=99, except for n=33 and 69. Brute-force 
> searching for a(33) shows that, if it exists, it must be > 10^11.  Does 

I hastily modified the little program I used to generate some sequences
such as A113856 or A097372.
Bugs apart, it seems to me that if a solution for the case n=33 exists,
then it must have 40 or more digits (so it's quite improbable it exists...).
Due to human-time-constraints, I did not use the shortcuts pointed out by Jim Nastos to 
restrict the range for the digits, and the search became quite slow at this point.
Probably with some modifications it is possible to prove that there are no solutions at 
all, in  reasonable time.
For the case n=69 I stopped the search at 30 digits with no solutions. Another hard case
seems to be n=111.

Since I'm a curious guy, I made a fast search for solutions (not only the least one)
with up to 21 digits for n<=300.

I did not find very large solutions. If the program is correct, the largest solutions
(10 or more digits) in the search range are:

{289,1857945889}
{298,1857945898}
{175,3197988689}
{166,3197988698}
{  5,3197988869}
{ 32,3197988896}
{104,3197988968}
{122,3197988986}
{241,5778863759}
{205,5778863795}
{ 43,5778863957}
{ 25,5778863975}
{223,7865675777}
{219,9937867749}
{174,9937867794}
{ 21,9937867947}
{  6,9937867974}
{291,179992689699}

giovanni