Summing Powers

For all positive integers \(1<n\leqslant k\), there exist non-negative single digit integers \(a_1,a_2,a_3,\ldots,a_n\) \((a_1\ne 0)\) which satisfy \[\overline{a_1a_2a_3\ldots a_n}=( a_1+ a_2+ a_3+\cdots+ a_n)^n.\]

Find the greatest value of \(k\).


This is one part of 1+1 is not = to 3.
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