Number of possible sets for given N How many possible valid collections are there for a given positive integer N given the following conditions:
All the sums from 1 to N should be possible to be made by selecting some of the integers. Also this has to be done in way such that if any integer from 1 to N can be made in more than one way by combining other selected integers then that set of integers is not valid.
For example, with N = 7,
The valid collections are:{1,1,1,1,1,1,1},{1,1,1,4},{1,2,2,2},{1,2,4}
Invalid collections are:
{1,1,1,2,2} because the sum adds up to 7 but 2 can be made by {1,1} and {2}, 3 can be made by {1,1,1} and {1,2}, 4 can be made by {1,1,2} and {2,2} and similarly 5, 6 and 7 can also be made in multiple ways using the same set.
{1,1,3,6} because all from 1 to 7 can be uniquely made but the sum is not 7 (its 11).

The term I would use is "multiset".  Note that your multiset must contain 1 (as this is the only way to get a sum of 1).   Suppose there are $r$ different values $a_1 = 1, \ldots, a_r$ in the multiset, with $k_j$ copies of $a_j$.  Then we must have $a_j = (k_{j-1}+1) a_{j-1}$ for $j = 2, \ldots, r$, and $N = (k_r + 1) a_r - 1$.  Working backwards, if $A(N)$ is the number of valid multisets summing to $N$, for each factorization $N+1 = ab$ where $a$ and $b$ are positive integers with $b > 1$ you can take $a_r = a$, $k_r = b - 1$, together with any valid multiset summing to $a-1$.  Thus $A(N) = \sum_{b | N+1, b > 1} A((N+1)/b - 1)$ for $N \ge 1$, with $A(0) = 1$.  We then have, if I programmed it right,  1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, 1, 3, 3, 8, 1, 8, 1, 8, 3 for $N$ from 1 to 20.  This matches OEIS sequence A002033, "Number of perfect partitions of n".
