# How many integer tuples $(n_1,…,n_k)$ satisfy $\frac{k-2}{2}\sum_{i=1}^{k} \frac{1}{n_i^2}<\sum_{i<j}\frac{1}{n_i n_j}$

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.

• There are many such tuples, and they won't have much difference between them. For example, $(100,100,100)$ or $(9990,9991,9992,\cdots ,10010)$. – didgogns Apr 7 '17 at 23:26
• If $k_i$ are really close to them, then the inequality is usually satisfied. – Levent Apr 7 '17 at 23:28
• Obviously there is infinite number of such tuples. But is this set countable? May be this is the quesion. – Smylic Apr 8 '17 at 1:09
• 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. – Levent Apr 8 '17 at 1:55