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Find the sum of all positive integers nnn for which 13+23+⋯+(2n)312+22+⋯+n2 \dfrac{1^3 + 2^3 + \cdots + (2n)^3 }{1^2 + 2^2 + \cdots + n^2} 12+22+⋯+n213+23+⋯+(2n)3 is also an integer.
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