1992 integra won't start

I have a 1992 Acura integra RS 1.8 L engine with 220,000 KM which intermittantly won't start. I turn the key and I get the red lights on the dash, no crank, no clicking noise. After turning the key a second or third time it starts right away. Sometimes it starts on the first try. I put in a brand new battery and no difference. The problem may have started after recently putting in a complete brand new distributor as the distubutor bearing was shot. All help greatly appreciated