Summing Powers

For all positive integers \(n\leqslant k\), there exists \(a_1,a_2,a_3,\ldots,a_n\) that satisfies the following: \[\overline{a_1 a_2 a_3\ldots a_n}=(a_1+a_2+a_3+\ldots+a_n)^n\] Find the greatest value of \(k\).

(Note: \(a_1,a_2,a_3,\ldots,a_n\) are digits of \(\overline{a_1a_2a_3\ldots a_n}\))

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