10   for N=1 to 3000

   20    if N=fnSod(2^N) then print N,2^N

   30   next

  999   end

 1000   fnSod(X)

 1010   local S$,Tot,I

 1015   Tot=0

 1020   S$=cutspc(str(X))

 1030   for I=1 to len(S$)

 1040     Tot=Tot+val(mid(S$,I,1))

 1050   next

 1060   return(Tot)

finds only

n     2^n

5    32

70   1180591620717411303424

Why no more?

There doesn't seem to be a specific reason ruling out further cases. The number of digits in the power should vary linearly with the power and should be about n multiplied by the common log of 2. The "typical" s.o.d. should also then be proportional to n. And the maximum possible sod would be 9*n*log(2), and as small as needed.

Perhaps it's just the great range and standard deviation in the sod that make it more and more unlikely to be a "hit" as the powers get larger.