smc

We can plug in 1, 8, and 49 to see which works. $1=1=1^2$ $1+2+3+\cdots +8=\frac{8\cdot 9}{2}=36=6^2$ $1+2+3+\cdots +49=\frac{49\cdot 50}{2}=1225=45^2$ All of these values produce a perfect square for $1+2+\cdots +K,$ so the answer is $\boxed{1,8,\text{ and }49}.$