Taking N=5, it's quite easy to see that every sum of {1,10,100,1000,10000} 
would be unique.

On the other hand, also for N=5, it's quite easy to see that the sums of
{2,4,6,8,10} would not all be unique.

So there is no such general formula.  That's because for there to be a general
formula, the number of distinct sums would have to be the same for all sets of 5
distinct numbers.  And as we see that is not the case. 

To esplain further with smaller set:

Suppose you are given the set of N = 3 numbers {1,2,3}.
Let's form all distinct sums by taking any number of
numbers and adding them:

1 = 1
2 = 2
3 = 3
1+2 = 3
1+3 = 4
2+3 = 5
1+2+3 = 6

The 3 appears as a sum twice, so there are 6 distinct sums.

Now suppose you are given the set of N = 3 numbers {1,2,4}.
Let's form all possible sums by taking any number of
numbers and adding them:

1 = 1
2 = 2
4 = 4
1+2 = 3
1+4 = 5
2+4 = 6
1+2+4 = 7

None of those sums appear twice, so there are 7 distinct sums.

No general formula can could give you both 6 and 7 when you 
substitute N=3 into it.

So there can be no such general formula.  It would depend on
how many different ways you could get the same sum by adding 
different combinations of the numbers.

Sorry.

Edwin