Patrick,
Well it seems that somewhere during the lifetime of your Granddad's Luger it was refurbished. The four numbers under the barrel originally should have been the same sn as on the frame, #7265, but 5??5 could have been that, not?
If this is very difficult to read, the pistol might have been polished once, and reblued. That explains the total absence of any proofmark.
The PSP P inside of the receiver are inspector marks, the meaning of the characters is not known. Many Lugers have them, and untill now no pattern is found here.
I am certainly not an expert, but I think that the only thing we can be certain about is that this is a DWM Luger, cal .30, short barrel, propably produced during the Weimar republic in Germany.
Not much of help, and certainly not what you wanted to know. If you take some time, and look further inside of this forum, you will learn that identification is only possible if there is something more than a serial number without a suffix.
Regards,