4. In the notebook, $n$ integers are written, ordered in descending order $a_{1}>a_{2}>\ldots>a_{n}$ and having a sum of 420. It is known that the $k-$th number written, $a_{k}$, except for the last one when $k=n$, is $(k+1)$ times smaller than the sum of all the other written numbers. Find the maximum number $n$ possible under these conditions. Find these numbers for the maximum possible $n$.
Answer: $n_{\max }=6 ; a_{1}=140, a_{2}=105, a_{3}=84, a_{4}=70, a_{5}=60$, $a_{6}=-39$.