It turns out to be a rather unremarkable number: 1729.
Given the time that the story was published, it is possible that Borges, who was fascinated by mathematics and always encoding maths concepts into his stories, had read the anecdote about the Hardy-Ramanujan number, and hid it purposefully in the story.
Edit: anyone interested in reading more about how Borges incorporated mathematics into his stories might find this book worthwhile
http://www.amazon.com/Unimaginable-Mathematics-Borges-Librar...