AN SF340'S LNDG GEAR IN-TRANSIT LIGHT REMAINED ON AFTER THE LNDG GEAR WERE UP AND THE GEAR DOOR LIGHTS WERE OUT. CREW LANDED AT THEIR TKOF ALTERNATE ARPT.

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DEPARTED ZZZ WITH WX CONDITIONS OF CALM WINDS; 2 MI VISIBILITY; CLOUDS AT 001 SCATTERED; OFF OF RWY 23. OUR TKOF ALTERNATE WAS ZZZ1. AS WE WERE INITIALLY CLBING OUT; THE FO SELECTED THE GEAR TO THE 'UP' POS WHILE I BEGAN A TURN TO THE TWR ASSIGNED HDG OF 300 DEGS. SHORTLY THEREAFTER; WE HAD NOTICED THAT THE GEAR IN-TRANSIT LIGHT DID NOT EXTINGUISH; HOWEVER; THE 3 GEAR LIGHTS DID EXTINGUISH AND THE HYD PUMP DID SHUT OFF. WE COMPLETED THE AFTER TKOF CHKLIST; AND NOTIFIED ATC THAT WE HAD AN ABNORMAL GEAR INDICATION. AT THIS TIME WE LEVELED OFF AT 4000 FT TWR ASSIGNED INITIAL ALT. I CALLED FOR THE QRH 'GEAR UP UNSAFE CONDITION' CHKLIST. WE SLOWED TO BELOW 150 KIAS; MADE SURE THE LNDG GEAR HANDLE WAS UP AND CHKED THE HYD INDICATORS WHICH ALL WERE NORMAL. THE MAIN HYD ACCUMULATOR PRESSURE WAS NOT LOW. WE CYCLED THE GEAR DOWN; 3 GEAR LIGHTS ILLUMINATED; THE IN-TRANSIT LIGHT EXTINGUISHED; AND THE HYD PUMP TURNED OFF. THEN WE PUT THE GEAR BACK UP TO COMPLETE THE RECYCLE STEP AND THE 3 GEAR LIGHTS EXTINGUISHED. HYD PUMP TURNED OFF; BUT THE IN-TRANSIT LIGHT REMAINED ILLUMINATED. AFTER THE RECYCLE STEP WAS COMPLETED AND OBSERVING WE HAD AN ABNORMAL INDICATION; WE PUT THE GEAR DOWN AND CONTACTED DISPATCH THROUGH ZZZ. AT THIS POINT IN THE FLT; I ASKED FOR A HDG FOR ZZZ1 (OUR TOA) AND CLBED TO 10000 FT WHILE CONTACTING CTR. STARTED TO LOSE CONNECTION WITH DISPATCH THROUGH ZZZ AND WAS RECONNECTED THROUGH A COMMERCIAL RADIO. I WAS NOW TALKING WITH DISPATCH AND MAINT CTL WHO WAS ALSO ON THE RADIO NOTIFYING MY DISPATCHER THAT I WAS ON A HDG TOWARDS ZZZ1. AFTER RELAYING THE SEQUENCE OF EVENTS THAT OCCURRED AND SOME Q&A WITH MAINT CTL; WE ALL DECIDED THE BEST CHOICE WAS TO PROCEED TO ZZZ1 (OUR TOA) BECAUSE WE DID NOT HAVE THE FUEL TO GO TO ZZZ2; CHOOSING NOT TO DECLARE AN EMER. WE DECIDED THIS BECAUSE THE GEAR WAS DOWN WITH 3 GEAR LIGHT INDICATIONS; HYD PUMP WAS NOT RUNNING; AND THE GEAR IN-TRANSIT LIGHT WAS EXTINGUISHED -- ALL NORMAL INDICATIONS FOR THE GEAR DOWN POS. AFTER ALL 3 PARTIES AGREED TO THE PLAN; I PROCEEDED TO AND LANDED AT ZZZ1 UNEVENTFULLY. PAX DEPLANED AND I CALLED DISPATCH TO RELAY THE MAINT LOG INFO. CONTRACT MAINT AND OUR MAINT CTL COMPLETED PAPERWORK. A FERRY PERMIT WAS ISSUED FOR US TO FLY ACFT FROM ZZZ1 TO ZZZ2 WITH THE GEAR EXTENDED IN NON ICING CONDITIONS BELOW 200 KIAS. WE LANDED IN ZZZ2 UNEVENTFULLY AND TOOK THE PLANE TO HANGAR.