Welcome to Android Central! Do you mean you want to use the Outlook app on your phone? Or do you mean that when you look at an email on your phone, it disappears from your Outlook 2013 on your computer?
Are you using a POP3 or IMAP server from Time Warner? Use the IMAP server if available, because with POP3, when the email is pushed to your phone, it usually gets deleted from the Time Warner server. This doesn't happen with IMAP.
What are my incoming and outgoing mail server addresses? | Time Warner Cable