It is well known that $\left(\sum_{k=1}^n k\right)^2 = \sum_{k=1}^n k^3.$
Now given $n$ integers $a_1,a_2,\ldots,a_n > 0$, is it possible to show that if
$\left(\sum_{k=1}^n a_k\right)^2 = \sum_{k=1}^n a_k^3,$ then $\{a_1,\ldots,a_n\} = \{1,\ldots,n\}$?
If it is false, for which $a_1,\ldots,a_n$ does this equality hold?
Thanks for your responses.