Growth rate of the eigenvalue of a rank-one matrix

For each $n$, let $x_n=(1,2,\dots,n)^T$ and $A_n=x_nx_n^T$, an $n\times n$ matrix with exactly one non-zero eigenvalue $\lambda_n$. For which exponent $k$ does $\lambda_n/n^k$ tend to a finite, non-zero limit?

$k=3$. $A_nx_n=x_n(x_n^Tx_n)$, so $\lambda_n=\|x_n\|^2=\sum_{j=1}^n j^2=\tfrac{n(n+1)(2n+1)}{6}\sim\tfrac{n^3}{3}$, giving limit $\tfrac13$ when $k=3$.