# A summoning of sums

For positive integers $$n$$ let $$S(n)$$ be the number of positive integer pairs $$(a,b)$$ such that $$a^{2} + a + n = b^{2}$$.

Let $$n_{k}$$ be the least positive integer for which $$S(n_{k}) = k$$.

Find $$\displaystyle\sum_{k=0}^{4} n_{k}$$.

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