The Answer is Not 196

Find the sum of all positive integers \(n\) such that

\[\sqrt{n}+\sqrt{n+60}=\sqrt{m}\]

for some non-square positive integer \(m\).


I'd just like to say that there are some really neat approaches and I encourage you to post a solution. I'd also really like to see a computer science based solution. Good luck! :D

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