B737 DIVERTS TO ANOTHER ARPT WHEN ACFT IS NOT PRESSURIZING DUE TO IMPROPERLY SEATED DOOR.

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DEPARTED ON SCHEDULE FROM RWY 1; WITH A TKOF WT OF APPROX 149K LBS. WITHIN THE FIRST 1500 FT OF THE CLB; IT WAS ESTABLISHED THAT THE ACFT WAS NOT PRESSURIZING. BLEED AIR AND PACK SWITCHES WERE VERIFIED TO BE IN THEIR NORMAL POS (ON) AND THE REAR OUTFLOW VALVE POS INDICATOR CONFIRMED THAT THE VALVE WAS CLOSED. NO OTHER ABNORMAL INDICATIONS WERE PRESENT IN THE FLT DECK. ALTHOUGH CLRED TO 11000 FT; WE REQUESTED TO LEVEL OFF AT 5000 FT. WE REMAINED CLR OF PROHIBITED AREA; WERE VECTORED TOWARDS ZZZ1'S AIRSPACE AND OPERATED THE ACFT AT CLEAN MANEUVERING SPD OF 230 KTS OR LESS. A CALL FROM THE AFT FLT ATTENDANT STATION REVEALED THAT THE R-2 SVC DOOR LEADING EDGE WAS PARTIALLY OPEN; THAT DAYLIGHT COULD BE SEEN THROUGH THE ENTIRE FORWARD EDGE OF THE DOOR BUT NOT ALONG AFT HINGED EDGE OF THE DOOR. THE DOOR LEVER WAS RPTED IN THE DOWN AND LOCKED POS. FLT DECK 'DOOR LIGHTS' WERE ALL EXTINGUISHED AND HAD BEEN SINCE PUSHBACK. THE QRH WAS CONSULTED FOR 'PRESSURIZATION' AND 'DOORS' ABNORMAL CHKLISTS AND DIRECTED US TO LAND AT THE 'NEAREST SUITABLE ARPT.' DISPATCH AND MAINT CTL WERE NOTIFIED AND CONSULTED WHILE WE WERE BEING VECTORED IN ZZZ1 AIRSPACE AT 5000 FT. IT WAS AGREED THAT WE WOULD LAND AT ZZZ1. THE FLT ATTENDANTS WERE NOTIFIED OF OUR INTENTION TO DIVERT TO ZZZ1 AND INSTRUCTED TO PREPARE THE CABIN FOR A 'NORMAL LNDG.' THE PAX WERE BRIEFED REGARDING OUR SITUATION AND NOTIFIED THAT WE WERE DIVERTING TO ZZZ1. WE NOTIFIED ATC OF OUR INTENTION TO LAND AT ZZZ1 AND DECLARED AN EMER IN ORDER TO OBTAIN PRIORITY HANDLING. WE REQUESTED TO HAVE EMER EQUIP STANDING BY AS A PRECAUTIONARY MEASURE IN CASE A LNDG ABNORMALITY DEVELOPED. WE BRIEFED AN ILS APCH TO RWY 1R WITH OVERWT LNDG CONSIDERATIONS. WE LANDED ON RWY 1R WITH FLAPS 40 DEGS; AT A LNDG WT OF APPROX 145K LBS; TOUCHED DOWN WITH A SINK RATE OF LESS THAN 300 FPM AND APPLIED BRAKING AT A GND SPD OF 80 KTS. WE CLRED THE RWY AND STOPPED TO LET THE EMER CREW ASSESS THE EXTERIOR CONDITION OF THE ACFT. THE ACFT WAS INSPECTED BY MECH (CONTRACT MAINT). THE R2 DOOR WAS EVALUATED AND IT WAS DETERMINED THAT IT HAD BEEN IMPROPERLY SEATED. THE R2 DOOR WAS PROPERLY SEATED AND CLOSED; THE ACFT WAS GND-PRESSURIZED AND INSPECTED PER PROCS. AN ADDITIONAL INSPECTION WAS DONE FOR THE OVERWT LNDG CONDITION. ALL TESTS AND EVALUATIONS WERE COMPLETED SATISFACTORILY AND THE ACFT LOGBOOK WAS SIGNED OFF. THE PAX WERE RE-BOARDED AND THE FLT PROCEEDED UNEVENTFULLY.