# For which minimal $k$ true is that ${4}^{k}\cdot n\leq \displaystyle\sum^{n}_{i=1}{a}_{i}^{k}\leq {5}^{k}\cdot n$, ${a}_{i}\in {1,2,3,4,5,6}$?

I've got the following inequality, which bounds Minkowski distance.
${4}^{k}\cdot n\leq \displaystyle\sum^{n}_{i=1}{a}_{i}^{k}\leq {5}^{k}\cdot n$
and values of ${a}_{i}\in {1,2,3,4,5,6}$
We know all values of ${a}_{i}$, the question is for which (minimal) integer $k$ above equation is satisfied ? Any necessary or sufficient conditions except obvious one that not for all $i$, $5\leq {a}_{i}$ ?

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Another obvious necessary condition is that the $a_i$ shouldn't all be $1$. But in general it seems to me that the minimal integer $k$ will depend in a complicated way on the $a_i$. I'd be surprised if there were a neat answer. – Gerry Myerson Mar 19 '13 at 23:11