What size beam should I use in my basement of my two story house, (3 story counting the basement) to span a distance of 13 feet? There are joists resting on an existing beam there now and these joists span 14 feet. The main beam right now has a post in the middle of this 13 foot span and I want to remove the post. i.e. the span of the existing beam is approx 6 ft and then a post and then 5 more feet.

You will need an engineer to determine this for you. It is a load bearing beam, we don't know what weight will be bearing on the structure. Typically, the beam companies will do the work for you.