The program finds the product of each set of digits that has zero to 3 of each digit 2 through 6 and zero through 2 of each digit 7 through 9.

The program tests each such product to see if there are enough of the above specified digts to form the product. If it does, it reports the product and the full digit set to get the product.  The product can also include up to three 1's. Except for a modification I made, the program would allow up to three zeros as well, but there are plenty of other digit combinations that padding with zeros is not that necessary.

clc

for d2=[1 2 4 8] 

    p=d2;

for d3=[1 3 9 27] 

    p2=p*d3;

for d4=[1 4 16 64] 

    p3=p2*d4;

for d5=[1 5 25 125] 

    p4=p3*d5;

for d6=[1 6 36 216] 

    p5=p4*d6;

for d7=[1 7 49] 

    p6=p5*d7;

for d8=[1 8 64] 

    p7=p6*d8;

for d9=[1 9 81] 

    p8=p7*d9;

    neededstr=char(string(p8));

   

    availstr=[repmat('2',1,log(d2)/log(2)) ...

              repmat('3',1,round(log(d3)/log(3))) ...

              repmat('4',1,log(d4)/log(4)) ...

              repmat('5',1,round(log(d5)/log(5))) ...

              repmat('6',1,round(log(d6)/log(6))) ...

              repmat('7',1,log(d7)/log(7)) ...

              repmat('8',1,log(d8)/log(8)) ...

              repmat('9',1,log(d9)/log(9)) ...

              '000' '111'];

    good=true;

     availhold=availstr;

    for i=1:length(neededstr)

      ix=strfind(availstr,neededstr(i));

      if isempty(ix)

        good=false;

        break

      end

      availstr(ix(1))='x';

    end

   

    if good

       if isempty(strfind(neededstr,'0'))

      disp(p8)

      disp(availhold(1:end-6))

      disp(' ')

       end

    end

    

end  

end   

end     

end   

end  

end   

end  

end

Finds necessary digits to produce such numbers. It specifically excludes embedded zeros.

The first non-trivial set found is

       18816

67788

1881677 is one such number.

Another example:

        1575 is the product of the digits in 5579, and thus also of 15759 (or 91575)

        

        

If we take examples of these and put them in the order of the products, the corresponding N's can be made to be in the same order by appropriate inclusions of the a given number of 1's at the end to make small numbers larger.

Other sets:

      139968

3346899

 

       11664

334669

 

   144

3344

 

     3483648

334467889

 

                1316818944

334466778899

 

   135

3335

 

    13395375

3335557799

 

    13716864

3334667789

 

     2

2

 

   128

288

 

   112

278

 

    12

26

 

        6912

26889

 

   672

2678

 

      186624

2466899

 

       16128

246678

 

       14112

246677

 

       14112

244779

 

      124416

2446899

trivial and non-trivial. This is a small portion of the list.

As a final example of the conversion, the set immediately above gives, for an example:

Product 124416 with included digits 2446899, as in 912441698 or say 91244169811.

Products that include zeros were explicitly excluded, as it seemed odd to ignore them in the multiplication, as in:

Product 11760 with included digits 56778, as in 71176085.