DURING A B737-700 PREFLIGHT THE FMC MESSAGE 'SINGLE FMC OPERATION' WAS CANCELED AND CONSIDERED TO BE A STARTUP NUISANCE. AFTER TAKEOFF THE CREW DISCOVERED FIRST THAT ONLY THE LEFT FMC WAS OPERABLE AND LATER THAT THE RIGHT FMC DID NOT HAVE CURRENT SOFTWARE.

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I WAS PERFORMING THE INITIAL PREFLT FOR THIS ORIGINATOR AT THE GATE. THERE WAS NO GND PWR AVAILABLE SO I STARTED THE APU AND PUT IT ON THE BUSSES. I CHKED THE FMC ACTIVE DATABASE AND IT WAS VALID AND BEGAN ALIGNING THE IRS. I RECEIVED A 'SINGLE FMC OP-' MESSAGE ON THE CDU; ALONG WITH A NUMBER OF OTHER MESSAGES AT START-UP. I'M NOT SURE OF THE OTHER MESSAGES; BUT THEY WERE STANDARD 'NUISANCE' MESSAGES LIKE 'CLOCK UPDATE.' I CLEARED THE OTHER MESSAGES QUICKLY AND CLEARED THE 'SINGLE FMC OP' MESSAGE AS WELL. THE DISPLAYS WERE STILL UNPOPULATED SINCE I HAD JUST PWRED UP THE AIRPLANE AND THE IRS'S WERE NOT ALIGNED YET. I ASSUMED THIS MESSAGE WAS DUE TO THE START-UP OF THE SYSTEM AND WOULD CLEAR ITSELF ONCE EVERYTHING ALIGNED. AFTER THE WALKAROUND; THE DISPLAYS APPEARED TO HAVE COME UP NORMALLY AND I HAD NO OTHER FMC MESSAGES; SO I ASSUMED EVERYTHING WAS OPERATING NORMALLY. I DID NOT INFORM THE CAPT OF ANY OF THIS SINCE IN MY MIND EVERYTHING CLEARED NORMALLY. I DID NOT NOTICE ANY UNUSUAL INDICATIONS OR DISPLAYS ON TAXI OUT. ON TKOF ROLL; I NOTICED THAT THE SPD MARKERS FOR THE V1 AND VROT SPDS WERE NOT PRESENT ON MY DISPLAY. I CALLED THE SPDS MANUALLY. ONCE SAFELY AIRBORNE WE ASSESSED THE PROB AND PERFORMED THE FMC FAILED CHKLIST IN THE QRH. THE R FMC WAS THE INOP FMC. THE L FMC OPERATED NORMAL AND THE REMINDER OF THE FLT WAS UNEVENTFUL. I REALIZE NOW THAT I MAY HAVE HAD AN AMBER 'MAP' INDICATION ON MY DISPLAY ON TAXI OUT. I ALSO NORMALLY CHK THAT MY V-SPEEDS ARE CORRECTLY SET BEFORE TKOF. OBVIOUSLY; I DID NOT DO THIS ON TAXI OUT. I SHOULD HAVE CAUGHT THESE INDICATIONS IN THE BLOCKS OR ON TAXI OUT. THE ROUTINE OF MULTIPLE 'NUISANCE' MESSAGES ON THE CDU REQUIRING DELETION TO CONTINUE; ESPECIALLY DURING START-UP CONTRIBUTED TO MY FAILURE. A SECONDARY ITEM OF NOTE: MAINT CHKED THE FMC AT OUR FIRST MAINT BASE. THEY DISCOVERED THAT NEW SOFTWARE WAS LOADED; BUT WAS ONLY LOADED ON THE L FMC AND DID NOT LOAD ON THE R FMC. I CHKED THE NAV DATA DATES ON MY ORIGINAL PREFLT AND THEY WERE VALID. THE NAV DATA VERSION NUMBER (NEW LOAD) HAD VALID DATES; AND THE OLD LOAD OF NUMBER HAD THE SAME VALID DATES. THE PRIMARY FAILURE WAS ME. I WILL PAY MORE ATTN TO MY DISPLAYS; PERSONAL FLOWS; AND FMC MESSAGES. ADDITIONAL LAYER OF PROTECTION COULD BE PROVIDED BY REDUCING THE NUMBER OF NUISANCE MESSAGES ON THE CDU. ALSO; EMPHASIZE THAT DURING OUR BEFORE PUSH CHKLIST WE SHOULD CHK THE V-SPEEDS ARE CORRECTLY DISPLAYED ON THE AIRSPD INDICTOR; NOT JUST CORRECTLY LOADED INTO THE CDU. THIS WOULD PROVIDE AN ADDITIONAL OPPORTUNITY TO CATCH THE FAILURE. ALSO; ANY NEW SOFTWARE CHANGE COULD BE AVAILABLE TO THE PLTS AND INCLUDED ONT HE WX PACKAGE OR RELEASE.