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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.

  • 2
    If you modify your question in such a way as to render already-written answers incorrect, then you invite undeserved down-votes on an innocent answerer. It might be better to indicate that you are making changes and/or additions to the question, so that readers will know that answers may be addressing an old version.2011-11-07

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