Coughing up nasty green phlem for several weeks?

Green sputum is an indication of a bacterial infection. You said you went to the Dr. and was given an antibiotic but apparently it didn't kill the bacteria, you need to be on another round of antibiotics different from the last prescription. Definitely have your Dr culture your sputum to find the right antibiotic to take care of this. Don't delay, hope you feel better soon. :)