There is no automated way to send out email to unconfirmed users. However, you could extract that information and send email using a different system. You could use the Discourse API, or possibly a Data Explorer query, to provide a list of email addresses of new users who signed up in the last week but have not confirmed.
I’d also note, there are integrated tools that provide this kind of task running, where you track a user’s path through your site. In this case the external tool would automate the process I described above.