You probably should demand a new phone number at no charge. I had that problem years ago upon receiving a new phone (receiving obscene calls late at night on the first night I got the line) and the phone co, after some pressure from me, changed the number. This is especially important on a cell phone where you have limited minutes and must pay as you go.
I can see this being a big problem on cell phones as people are more likely to "dispose" of them than their home landline phone. For example, a father might pull the plug on a teenager's line for a while. That means the numbers will get used much more frequently. I would not want a cell phone number previously assigned to a teenager.[Ohmygawd that would be, like, SO weird, you know, to have this old guy answering your cell phone, you know? Like my friend Jennifer? She is SUCH a flirt and she, like, would REALLY flirt with guys she doesn't even know who call her, like, what is that girl thinking? And at Lindsay's party she threw such a fit because Lindsay was using Jeremy's cell phone and YOU KNOW Jennifer really likes Jeremy but just won't show it because Jeremy isn't cool enough for her and she goes out with Trevor instead even though Trevor is really mean to her and cheats on her ALL THE TIME and she's so stupid she doesn't even see it right in front of her, like when Trevor was making out with Morgan -- I swear this is true -- at Amanda's house right in the living room in front of everybody, Morgan has like no shame and is the WORST slut in school, worse even than Jennifer's older sister, remember her, she got sent home for wearing her blouse too tight and her parents came back and, like, yelled at Mr. Green the principal to mind his own business and then he got REALLY mad told them she was three months pregnant which was supposed to be a big secret but everybody knew it ...]
Since you are getting many calls for an old number it wouldn't pay to track each one down. In some cases old numbers are reassigned too quickly. One way to avoid that problem is to request a number in a new exchange, or better still, a new area code which your area might have.
If you are getting repeated abusive anonymous calls, there is Call Trace. *57 (1157 for rotary users). This cannot be blocked. The phone company gets the number in a special unit. You may have to pay a fee for this. For repeated abuse calls, the phone company or police will deal with it.
Pat mentioned some other *dial options to screen calls, but note usage varies in some areas and there may be a fee to use them.