Battery question

I have a 1999 Dodge Intrepid with the larger available engine. It has 110K miles on it. I am not sure of the age of the battery as I bought the car used.

On occasion when I turn the key, nothing happens. No clicking of the starter or anything. It seems totally dead except that all electrical things(lights, radio, etc) seem to be working fine. If I turn the key enough times (sometimes 20 or so times), it will eventually start like normal. Once warm it does not seem to do it as frequently. It also will go several weeks at a time without incident and then start happening again. Do you think that this is a battery issue or a starter issue?

thanks

Greg