Question Regarding PS1 Classics On PS3/PSP

I have a Playstation 3 that I'm pretty sure is out of gameshares and I'm planning on buying a new PSP. I know I probably can't use my currently purchased PS1 classics on the PSP, so would I be able to purchase PS1 classics through the PSP and use those on both the PS3 & PSP?

Thanks

Oh, that's awesome. Thank you for the response.

Since the accounts are the same on both your PS3 and PSP you should just be able to go to the recent downloads on either the psp or ps3 and download it .