Handsome birds aren't they. They are quite partial to whole peanuts and in the autumn you will often see them caching them in the lawn for later in the winter. Whether they ever find them again I don't know..
They often don't find them again. Jays will also cache acorns, but many of the ones that they aren't able to relocate end up germinating and sprouting new trees.
Perhaps should mention that loose whole peanuts should only be put out when there are not young birds about, the story being that small chicks like Blue Tits can choke on whole peanuts although feeding kibbled peanuts is alright at this time of year, apparently.