Where the string buzzes on the neck is probably the most important part of diagnosing your problem. In fact, po0sition of the buzz will generally tell you exactly where the problem lies.
The fact that it doesn't buzz when laying flat idicates that neck droop is ti cause of yourv problem. If you will notice, the strings that are working properly will probably change pitch ever so slightly when you change the bass from a flat position to the normal playing position. The change is ever so slight and will probably only be noticable when checking with your strobe tuner. This indicates that there is really a measurablr change between flat and normal playing position. Gravity is probably the culprit, dictating that a correction in the set-up must be done to compensate.
If all the other strings are normal, simply raising the A string with with the bridge saddle hiegth will probably correct the prob.
Less likely but not unheard of, you may have put on a defective new string.
Your problem usually pops up whe one has adjusted the bass for the absolute lowest possible action.
Gary willis has a website that does a pretty good job of explaining the diagnosis procedure. A quick search will get you to his website.
Good luck and be patient.
Pkr2+