6、Given $\left[\prod_{k=1}^{n} \frac{(2 k+1)^{4}+(2 k+1)^{2}+1}{(2 k)^{4}+(2 k)^{2}+1}\right]=44$. Then $n=$ $\qquad$, $[x]$ denotes the greatest integer not exceeding the real number $x$.

 untranslated part:
已知 $\left[\prod_{k=1}^{n} \frac{(2 k+1)^{4}+(2 k+1)^{2}+1}{(2 k)^{4}+(2 k)^{2}+1}\right]=44$. 则 $n=$ $\qquad$ , $[x]$ 表示不超过实数 $x$ 的最大整数.

Corrected and translated part:
Given $\left[\prod_{k=1}^{n} \frac{(2 k+1)^{4}+(2 k+1)^{2}+1}{(2 k)^{4}+(2 k)^{2}+1}\right]=44$. Then $n=$ $\qquad$, $[x]$ denotes the greatest integer not exceeding the real number $x$.
6. 5

Analysis: Given $n^{4}+n^{2}+1=\left(n^{2}+n+1\right)\left(n^{2}-n+1\right)$, and $n^{2}+n+1=(n+1)^{2}-(n+1)+1$, $n^{2}-n+1=(n-1)^{2}+(n-1)+1$, hence $\prod_{k=1}^{n} \frac{(2 k+1)^{4}+(2 k+1)^{2}+1}{(2 k)^{4}+(2 k)^{2}+1}=\frac{(2 n+1)^{2}+(2 n+1)+1}{3}$, solving for $n$ yields $n=5$.