A025591 and subset sums

[T. D. Noe]

>> > If my memory is not mistaken, sequence A025591 is related to the following
>> > paper :
>> >
>> > Robert A. Proctor
>> > Solution of two difficult combinatorial problems with linear algebra,
>> > American Mathematical Monthly 89, 721-734.
>>
>> Here's the abstract. I don't see the connection.
>
>A025591(n) is the maximal number of subsets of $\{1,2,\dots,n\}$ that
>share the same sum:
>
><<DiscreteMath`Combinatorica`
>Max/@Map[Length,Split/@Sort/@((Plus@@@Subsets[#])&/@Range[10]),{2}]
>{1,1,2,2,3,5,8,14,23,40}

The same idea can be used on subsets of other sequences:

A025591 uses integers {1,2,3,...n}
A083309 uses odd primes
A083527 uses squares of integers

So this operation transforms one sequence into another.  What is the
inverse transform?

Tony