Conjecture: There is no $n\neq3$ such that $\arctan(1)+\arctan(2)+\cdots+\arctan(n)=\frac{k\pi}{2}$, with $k$ integer.

Question

Fact: $\arctan (1) + \arctan (2) + \arctan (3) = \pi$.

Conjecture: there is no $n \neq 3$ such that $\arctan (1) + \arctan (2) + \cdots + \arctan (n) = \frac{k\pi}{2}$, with $k$ integer.

I know a stronger result than conjecture is: The product $(1 + 1^2) (1 + 2^2) \cdots (1 + n^2)$ is just a perfect square for $n = 3$