One for All; All for One

Algebra Level 4

Let \(g\) be a function defined over all positive integers such that \(g(1) = 1\) and \(\displaystyle\sum_{k=1}^{n} g(k) = n^{2}g(n)\) for \(n \ge 2.\)

If \(2015 \cdot g(2015) = \dfrac{a}{b},\) where \(a\) and \(b\) are positive coprime integers, then find \(a + b.\)

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