B757-200 FLT CREW HAS LOSS OF PWR IN THE R ENG DURING CRUISE FLT AT FL350.

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CRUISING ALONG; BEGAN TO LOSE PWR ON R ENG. OUR RTE WITH MINOR DEVS HAD UNTIL THIS POINT KEPT US CLR OF ALL WX AND CLOUDS. WE NOTICED WX ACTIVITY AROUND THE MOUNTAINS TO THE W OF US. EARLIER IN THE FLT WE ESTABLISHED COM WITH ANOTHER FLT PROCEEDING ALONG PART OF OUR RTE ABOUT 10 MINS AHEAD OF US. LATER; WE WERE CLOSE ENOUGH TO HEAR THE XMISSIONS. PERFORMANCE OF OUR HF RADIOS MARGINAL THIS NIGHT. OUTSIDE OF READING THE ENG GAUGES THERE WERE NO EICAS OR STATUS MESSAGES TO INDICATE WHAT TYPE OF ENG PROB MIGHT BE OCCURRING. R ENG EPR PEGGED HIGH EGT HUNG AT 670 DEGS C MATCHED SEIS INDICATOR. N1; N2; N3; FF ALL DECREASING. TURNED OFF THE AUTOTHROTTLES. TURNING ON THE R ENG ANTI-ICE CAUSED A HVY VIBRATION IN THE RUDDER PEDALS AND FLT DECK FLOOR. NOT SURE AT THE TIME JUST WHERE WE COULD HAVE PICKED UP ANY ICE; WE THOUGHT THE VIBRATION MIGHT BE ICE SHEDDING. WE TOOK MANY THINGS INTO CONSIDERATION INCLUDING FUEL CONTAMINATION. AS OUR AIRSPD BEGAN TO DECREASE; I DIRECTED THE FO TO REQUEST FROM CTL DESCENT TO FL310. CTL INITIALLY DENIED OUR REQUEST WITH AN UNABLE REQUEST FOR FL310 RESPONSE. AT THIS POINT WE HAD SLOWED FROM MACH .81 DECELERATING THROUGH MACH .77. I DIRECTED THE FO TO DECLARE AN EMER WITH MAYDAY X3. WITH OUR NEW STATUS; CTL GAVE US CLRNC IMMEDIATELY TO FL310 WITHOUT REF TO ANY CONFLICTING AIR TFC. ONCE LEVEL AT FL310 AFTER ABOUT 4 MINS; THE R ENG PWR STABILIZED AT A REASONABLE PWR SETTING. HOWEVER; WE STILL HAD VIBRATION IN THE FLT DECK FLOOR BOARDS AND RUDDER PEDDLES WITH NO ADDITIONAL SUPPORTING INDICATIONS FROM OUR ENG INSTS. THE VIBRATION MONITOR OF THE R ENG CONTINUOUSLY SHOWED N1 VIBRATION LEVEL 1.3. STILL IN CLR AIR; WE DECIDED THE R ENG SHOULD HAVE BEEN WELL CLRED OF ANY ICE AND WE SHUT THE ENG ANTI-ICE OFF. THE VIBRATION WENT AWAY. TURNING THE ENG ANTI-ICE BACK ON BROUGHT THE HULL VIBRATION BACK. THE FO AND I CONCLUDED THE BLEED OFF OF THE R ENG FOR ANTI-ICE WAS SOMEHOW DISTURBING THE NORMAL FLOW OF AIR THROUGH THE ENG CAUSING VIBRATION. OTHER COMPANY FLTS MADE PATCHES TO DISPATCH. BTWN THESE 2 ACFT WE WERE ABLE TO GET INFO TO AND FROM DISPATCH AND MAINT. ACCORDING TO MAINT; OUR ACFT HAD A HISTORY WITH LOSS OF ENG PWR ON THE R ENG FL350 AND ABOVE. IN ALL CASES; THE R ENG STABILIZED WITH A REASONABLE PWR SETTING AT A LOWER THAN FL350 CRUISE ALT. DISPATCH OUTLINED NUMEROUS DIVERSION ARPTS WHERE WE HAD PARTS AND OPS. THE RELIEF PLT MONITORED THE NEAREST EMER ARPTS IN CASE A LNDG ASAP SHOULD BECOME NECESSARY. WITH ALL THE INFO; HVY WT; WX AND POSSIBILITY NO R ENG; NO R ENG ANTI-ICE; ENG HISTORY AND FOR THE TIME BEING STABILIZED PWR ON THE R ENG. I ELECTED TO MAINTAIN FL310; STAY AWAY FROM WX; MOUNTAINOUS TERRAIN INCLUDING OUR PLANNED DEST. AND DIVERT.