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How many integer tuples $(n_1,...,n_k), k>3$ where $M<n_i<N$ satisfy the above equation?

I am trying to find the number of solutions to the equation $$n\cdot\sum_{i=1}^{n+1} 1/k_i^2 < (\sum_{i=1}^{n+1} 1/k_i)^2.$$ in order to solve a problem about tangent circles. I get the equation using Descartes' Theorem.

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    $\begingroup$ There are many such tuples, and they won't have much difference between them. For example, $(100,100,100)$ or $(9990,9991,9992,\cdots ,10010)$. $\endgroup$ – didgogns Apr 7 '17 at 23:26
  • $\begingroup$ If $k_i$ are really close to them, then the inequality is usually satisfied. $\endgroup$ – Levent Apr 7 '17 at 23:28
  • $\begingroup$ Obviously there is infinite number of such tuples. But is this set countable? May be this is the quesion. $\endgroup$ – Smylic Apr 8 '17 at 1:09
  • $\begingroup$ There are finitely many tuples since $M<n_i<N$ for all $i$. Even if this is not the case, the number of tuples must be countable since these tuples are elements of $\mathbb{Z}^k$ which is a countable set. $\endgroup$ – Levent Apr 8 '17 at 1:55

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