B757 FLT CREW NOTICES LOW ENG OIL QUANTITY SHORTLY AFTER LEVELOFF AT FL350; ENG IS SECURED AND FLIGHT DIVERTS TO NEAREST SUITABLE ARPT.

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ACFT XYZ WAS DISPATCHED WITH THE R ENG EEC AND L AIR CONDITIONING PACK INOPERATIVE. PREFLT; ENG START; AND TAXI WERE ACCOMPLISHED AS OUTLINED IN THE OPERATING MANUALS WITH NO ABNORMALITIES NOTED. FO WAS FLYING. TKOF AND CLB TO FL350 WERE COMPLETED WITHOUT ANY ADVERSE INDICATIONS. APPROX 30 MINS INTO THE FLT AT FL350 FO RPTED SEEING A FLICKERING 'OIL PRESS' LIGHT. UPON REVIEW OF THE EICAS SCREENS I NOTED THAT THE OIL QUANTITY FOR THE R ENG WAS APCHING ZERO. EICAS OIL PRESSURE INDICATION FOR THE R ENG WAS FLUCTUATING AND OIL TEMP INDICATION WAS INCREASING. BOTH INDICATIONS WERE CONSISTENT WITH A LOSS OF OIL QUANTITY. WE COMPLETED THE ENG OIL QUANTITY LOW AND ENG OIL PRESS PROCS IN THE QRH WHICH LED US TO THE ENG FAILURE OR PRECAUTIONARY SHUTDOWN CHKLIST. FO FLEW WHILE I REQUESTED A LOWER ALT AND ADVISED ATC THAT WE WERE SHUTTING DOWN AN ENG AND WOULD BE DIVERTING TO ZZZ. WE DECLARED AN EMER AND ATC APPROVED THE L TURN DIRECT TO VOR THEN DIRECT ZZZ WITH AN INITIAL DSCNT TO FL290. I CONTACTED FLT ATTENDANT #1 AND ADVISED HER OF OUR PRECAUTIONARY SHUTDOWN AND TO PREPARE FOR LNDG AT ZZZ. I ADVISED HER THAT WE WOULD BE LNDG IN APPROX 20 MINS. I ADVISED FLT ATTENDANT #1 THAT WE PLANNED A NORMAL LNDG WITH NO PLANNED EVAC. I MADE AN ANNOUNCEMENT TO THE PAX EXPLAINING OUR SITUATION AND ADVISED THEM THAT WE WOULD BE STOPPING ON THE RWY AFTER LNDG AND AFTER AN EXTERNAL INSPECTION BY THE EMER PERSONNEL WE WOULD BE TAXIING TO THE GATE. I OPENED THE ISOLATION VALVE IN ORDER TO MAINTAIN ACFT PRESSURIZATION BECAUSE THE L PACK WAS INOPERATIVE AND I NEEDED TO RESTORE AIRFLOW TO THE R PACK. I ASKED ATC TO ADVISE DISPATCH OF OUR INTENT TO DIVERT TO ZZZ IN CASE I WAS UNABLE TO CONTACT THEM IN A TIMELY MANNER. WE RECEIVED VECTORS FOR THE ILS XX AT ZZZ WITH CALM WINDS AND CAVU WX. WE COMPLETED THE 1 ENG APCH AND GAR PROC. AN UNEVENTFUL APCH AND LNDG WERE COMPLETED. FROM THE TIME WE SECURED THE R ENG UNTIL WE WERE ON GND IN ZZZ WAS APPROX 25 MINS. AARF CHKED THE ACFT AND RPTED OIL DRIPPING FROM THE R ENG. THEY THEN MOVED OFF TO THE SIDE OF THE RWY ALLOWING US TO TAXI TO THE GATE. DUE TO THE UNCERTAINTY OF THE LENGTH OF OUR DELAY THE ZZZ STATION PERSONNEL REQUESTED THAT THE PAX REMAIN ON BOARD. WE AGREED AND THE FLT ATTENDANT COMMENCED SERVING BREAKFAST IN FIRST CLASS AND A BEVERAGE SVC IN THE MAIN CABIN. THE CABIN TEAM DID AN EXEMPLARY JOB KEEPING THE PAX COMFORTABLE AND INFORMED. THE R ENG WAS CLEANED AND THE MAINT PERSONNEL RPTED THAT THE OIL FILLER CAP APPARENTLY WAS NOT COMPLETELY SECURED. THE OIL QUANTITY WAS REPLENISHED AND THE ENG MOTORED FOR 30 SECONDS. THEN MAINT RECHKED THE ENG FOR ANY LEAKAGE AND RECHKED THE OIL QUANTITY. AFTER REFUELING AND PREFLT INSPECTIONS WE DEPARTED AND FLEW UNEVENTFULLY TO ZZZ1.