Call a positive integer n "favorable" if there is a set of n distinct positive integers whose reciprocals' sum adds to 1.
How many unfavorable numbers are there?
If 1/a+1/b+...+1/y+1/z=1, and 1<a<b<...<y<z, then 1/a+1/b+...+1/y+1/(z+1)+1/z(z+1)=1, and 1<a<b<....<y<z+1<z(z+1), so if N is favorable, it follows that N+1 is also favorable.
Since 1/2+1/3+1/6=1, all numbers from 3 onwards are favorable. Add that N=1 is favorable, and N=2 isn't, and you get the complete answer.