A summoning of sums

For positive integers nn let S(n)S(n) be the number of positive integer pairs (a,b)(a,b) such that a2+a+n=b2a^{2} + a + n = b^{2}.

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

Find k=04nk\displaystyle\sum_{k=0}^{4} n_{k}.

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