IR-TST 2019 Problem 12

For any positive integer $n$ , define the subset $S_n$ of natural numbers as follow $$S_n = \left\{x^2+ny^2 : x,y \in \mathbb{Z} \right\}.$$ Find all positive integers $n$ such that there exists an element of $S_n$ which doesn't belong to any of the sets $S_1, S_2,\dots,S_{n-1}$ .

Proposed by Yahya Motevassel