What was found
NEW NOSE TIRE INSTALLED BEFORE FLIGHT. UPON TAKEOFF AND SELECTING GEAR UP HEARD LOUD NOSE WHEEL SPINDOWN AND FELT ASSOCIATED VIBRATION. NOISE CONTINUED FOR SEVERAL SECONDS. AFTER SELECTING GEAR UP THE RED GEAR UNLOCK LIGHT REMAINED ON. ALL CIRCUIT BREAKERS WERE IN. SELECTED GEAR DOWN AND THE RED GEAR UNLOCK LIGHT EXTINGUISHED AND OBSERVED NORMAL GEAR DOWN AND LOCKED INDICATIONS. PNEUMATIC BOTTLE WAS NOT BLOWN. NO OVERWEIGHT LANDING. ON POSTFLIGHT NOTICED PAINT MISSING ON RIGHT SIDE OF NOSE LANDING GEAR FORK, AND GREASE PRESENT ALONG NOSE LANDING GEAR OUTER STRUT COLLAR. CORRECTIVE ACTION: REPLACED WHEEL/TIRE ASSY. OPS CHECK GOOD. IAW MM 32-41-01.