Okay.. then call Apple Tech Support and ask for their help as this is not normal behavior... The serial code was only needed with versions of Logic Pro installed from DVDs... or being upgraded from earlier versions. Since the App Store version was released, serials were no longer used...
I have downloaded / installed Logic Pro many times from the App Store and it has never asked for a serial... so Ihave to assume something weird is going on here and only a call to Apple will be able to resolve it..
Good luck.... and sorry I cannot help you further...