# Searching for the identity of CS sets

I apologize in advance if this question is too surface level.

I have heard recently about such mathematical objects as "CS sets". They are defined as sets of natural numbers $a_1,a_2,\ldots,a_n$ such that $$\sum\limits_{i=1}^n a_i^3 = \left(\sum\limits_{i=1}^n a_i\right)^2.$$ CS sets are denoted by the notation $\langle a_1,a_2,\ldots, a_n\rangle$ using angled brackets instead of curly brackets, to distinguish them from regular sets. In fact, these things are not actually sets, contrary to their name: they allow for duplicate elements in them, unlike sets. For example, $\langle 3,3,3\rangle$ is a valid CS set, as $$3^3+3^3+3^3 = 81 = (3+3+3)^2.$$

I have heard that these objects have been defined long ago and here has been a lot of literature on the subject. Oddly, however, googling has not yielded any results at all, and I can't seem to find related articles anywhere online.

I want to ask where to look online for these objects (what terms to search, on what website), and specifically the names of significant researchers in the field. If possible, I also want the specific subfield of math this belongs in (under number theory). Also, "CS" sounds awfully much like an abbreviation, so I would also like what it stands for. Links to some important articles, papers or books on the topic are also really appreciated. Many thanks in advance!

• I dk. Except to say that if $A_n= \{a_1,..., a_n\}$ is the set of the first $n$ natural numbers then $A_n$ is a CS set because $\sum_{j=1}^nj^3=(\sum_{j=1}^nj)^2=(n(n+1)/2)^2.$ – DanielWainfleet Sep 1 '18 at 20:26
• Well obviously yes. Some other nice results have also been proven, like there doesn't exist a CS set with two '3's and all other elements distinct. – YiFan Sep 2 '18 at 2:24