The short story: As mentioned below, the smallest (and only) factoritwo is 3679464.

for n=1000000:3000000

   s=num2str(n);

   tot=0;

   for i=1:length(s)

      tot=tot+factorial(str2double(s(i))+1); 

   end

   if tot==n

      disp(n); 

   end

end

found nothing, so, figuring it needed to go higher, I switched strategies. Instead of testing each number, I'd test combinations of digits. That would go faster, as 213457 and 719 others would be in the batch considered as 123457 where the total 2+6+24+120+720+40320 = 41192 does not consist of those digits. The sum of the incremented factorials of those digits would be tested to see if it matched the numbers. I'd allow for eight-digit numbers:

for i=1:10

   f(i)=factorial(i); 

end

for zeros=0:8

    z=repmat('0',1,zeros);

    for ones=0:8-zeros

        o=repmat('1',1,ones);

        for twos=0:8-zeros-ones

            t=repmat('2',1,twos);

            for threes=0:8-zeros-ones-twos

                th=repmat('3',1,threes);

                for fours=0:8-zeros-ones-twos-threes

                    fo=repmat('4',1,fours);

 for fives=0:8-zeros-ones-twos-threes-fours

    fi=repmat('5',1,fives) ;

    for sixes=0:8-zeros-ones-twos-threes-fours-fives

        sx=repmat('6',1,sixes);

        for sevens=0:8-zeros-ones-twos-threes-fours-fives-sixes

            sv=repmat('7',1,sevens);

            for eights=0:8-zeros-ones-twos-threes-fours-fives-sixes-sevens

                e=repmat('8',1,eights);

                for nines=0:8-zeros-ones-twos-threes-fours-fives-sixes-sevens-eights

                    ni=repmat('9',1,nines);

                    dig=[z o t th fo fi sx sv e ni];

                    tot=0;

                    for i=1:length(dig)

                        tot=tot+f(str2double(dig(i))+1);

                    end

                    tt=char(string(tot));

                    if isequal(sort(tt),sort(dig))

                       disp(tt); 

                    end

                end

            end

        end

    end

 end

                end

            end

        end

    end

end

finds

3679464

     24
   5040
  40320
3628800
    120
   5040
    120
-------    
3679464 

found when testing the combination 3, 4, 4, 6, 6, 7, 9.

    

I had really only needed to go another million in the original program, which would have been quicker than writing a new, faster, program.

Also, I think 3679464 is the only such number. All 8-and-fewer-digit possibilities have been tested (that's one advantage of the new program). The most that a 9-digit number could add to would be 9*(10!) =  32659200, which has only 8 digits, so higher than 8 digits would be impossible..