Let $\sigma(X)=\sum_{x\in X}x$ Construct it inductively as follows. Set $x_1=1$. If the first $n$ numbers of the set are $x_1,...,x_n$, then for any subset $A_i\subseteq \{1,...,n\}$ pick a different prime $p_i$, and pick $x_{n+1}$ so that $x_i+\sigma(A_i)\equiv p_i\pmod{p_i^2}$ for all $i=1,...,2^n$. By CRT this can be done, and so $\sigma(\{x_{n+1}\}\cup A_i\})$ can not be a perfect square, nor a perfect power, since the $v_{p_i}$ of this number is $1$.