For which n is it possible to split all the integers 1, 2, 3, ..., n into two non-empty disjoint sets such that the product of the sum of the elements in one set and that of those in the other is a perfect square?
It is easy to see that any two-way split S=A+B of the sum S=1+..+n=n(n+1)/2 can be achieved by forming two disjoint subsets. Let A=ac,B=bc with a,b relatively prime AB is a perfect square precisely if a and b are. Therefore
n has the desird property if and only if S has a divisor that is a sum of two squares.
this will be the case
iff S has at least one prime factor that is not congruent 3 mod 4.
The theorem doesn't rule out even powers of primes of the form 4k+3 but that is only because it allows sums where one term is zero.