I personally use 6.20 PRO b-10 perma patched on my PSP 3000.
Download 6.20 Official firmware here: :: zload.net ::
and install this: 620PRO-B10.fix1.rar - procfw - 6.20 PRO-B10.fix1 - X-Generation Aftermarket Firmware for Playstation Portable
Just follow these instructions, and it should work.
1. Put the contents of PSP/GAME into your PSP/GAME folder on the memory stick.
2. Run the PROUPDATE and follow the on screen instructions.
3. Run the 620PRO_Permanent and follow the on screen instructions. MAKE SURE YOU TEST IT.
4. If it works after you test it, then run the 620PRO_Permanent again, and don't test it.
5. You are now on a Permanent patched custom firmware.
EDIT: before you run the official firmware update, make sure your battery is charged.