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4001 Pickups

So far there are four best known manufacturers:
  • Rickenbacker's original Hi-Gains and Toasters, which, IIRC, are single-coils, and HB1 which is a humbucker. Most can fit in either position, I think, though Ric sells separate neck and bridge units.
  • Seymour Duncan's SRB1n and SRB1b, which are both humbuckers. SRB1n is a neck unit, SRB1b is a bridge unit.
  • Bartolini's 6RT, 6RC and 6JH which are a neck split-coil, a bridge full-width humbucker and a special mute-compartment pickup which gives extra "zing", mounted instead of the bridge foam mute and usually wired in series with the 6RT. All humbuckers.
  • Kent Armstrong's got the SRK1F and SRK1R pickups for front (neck) and rear (bridge) positions, intended that way due to differing amounts of wire on each coil. Supposedly humbucking.

Those are the most common ones - I'm sure there's many talented winders who dare wind their own Ric-oid pickups. The last time someone made a Ric-alike bass, he used a Kent Armstrong SRK1F and a Seymour Duncan SRB1b.
 
Your best bet is 4003 high gain pickups or used 4001 high gain pickups on ebay. Your 4001 came from the factory with 8k ohm single coil pickups. The modern 4003 pickups are the same basic design except they are 11k ohms. The will be slightly louder and fuller than the 4001 pickups.