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I just got a used PSP with CFW 3.90 ME, updated it to OFW 6.61 and activated it for my account.
To speed up the downloads from the PSN Store I downloaded some PSP games to my PS3 (OFW 4.82).
But when trying to copy those games to the PSP or pair the PSP for Remote Play, then my PS3 says that there are no PSP systems.
Both systems are logged into PSN.
The PSP is in USB mode, the progress bar is shown for some seconds and then is gone as if the PSP is connected.
Connection from the PSP to a PC is working well, that's how I copied the OFW 6.61 to it.

Any ideas?

Lent a PSP-3004 and was able to connect this PSP and my PS3.
Therefore it is really an issue with my PSP and not the PS3 or usb cables.
Next is to find out if it is software-wise (e.g. due to the previous CFW) or hardware-wise.

So how can I recover OFW 6.61 100% correctly on my PSP?

To make sure that I have a 100% clean OFW I created a Pandora setup (service mode battery via serial hack plus magic memory stick), then installed OFW 5.00 via the IPL load and it did indeed clear flash0 to flash3. After that I installed OFW 6.60, but still no connection possible

At the end it turned out that some IDStorage entries were corrupted and I could fixed them with KeyCleaner: entries 0x42-0x47 only.
Did not "patch" 0x05 as I didn't had any update issues.