Let $a$ and $d$ be the first term and the common difference of the arithmetic sequence.
$\begin{array}{rcl}
T(11) & = & 83 \\
a+10d & = & 83 \ \ldots \unicode{x2460}
\end{array}$
Also, we have
$\begin{array}{rcl}
T(25)+T(30) & = & 463 \\
a+24d+a+29d & = & 463 \\
2a+53d & = & 463 \ \ldots \unicode{x2461}
\end{array}$
$\unicode{x2461}-2\times\unicode{x2460}$, we have
$\begin{array}{rcl}
33d & = & 297 \\
d & = & 9
\end{array}$
Sub. $d=9$ into $\unicode{x2460}$, we have
$\begin{array}{rcl}
a+10(9) & = & 83 \\
a & = & -7
\end{array}$
Therefore, the first term and the common difference of the arithmetic sequence are $-7$ and $9$ respectively.
Hence, we have
$\begin{array}{rcl}
T(1)+T(2)+\cdots+T(k) & > & 4\times 10^5 \\
\dfrac{k}{2}[2(-7)+(k-1)(9)] & > & 4 \times 10^5 \\
9k^2-23k & > & 8\times 10^5\\
9k^2-23k-8\times10^5 & > & 0
\end{array}$
$\therefore k<-296\ 867\ 357\ 4$ or $k>299.422\ 912\ 9$.
Hence, the lease value of $k$ is $300$.
