By checking digital roots
N 2^N
0 1
1 2
2 4
3 8
4 7
5 5
6 1
7 2
8 4
9 8
10 7
11 5
and noting the cycle in the 2^N column it is clear that most numbers don't have a chance.
In fact only if N is of the form 18a+5 will the digital roots of N and 2^5 be the same (5).
I could only check 5, 23, 41 on my calculator and only N=5 works.
It's possible, but unlikely there are other solutions, but the number of digits of 2^N grows as log(2) and the sum of digits grows as 4.5log(2)=1.35 it would be surprising to see another solution.

Posted by Jer
on 20160527 11:40:18 