jmc

First of all, we see that every term has a remainder of $1$ when divided by $6.$ Now, we just need to find how many terms there are. The nth term can be given by the expression $6n-5.$ Therefore, we set $259=6n-5$ to find $n=44,$ thus there are $44$ terms in our sum. Thus, the remainder of the sum is the same as the remainder of $44$ when divided by $6,$ which is $\boxed{2}.$