When viewing [[Special:Search]] the focus is always on the search input field. This is useful when you are going to start a new search. But when you are viewing results for a previous search it is annoying that you can not start pagedowning without having to unfocus the search input field.
I suggest the focus is only when viewing [[Special:Search]] without any search terms (and maybe if the search terms gives no results). Not when viewing search results like [[Special:Search/search+word]].