If you've purchased an older version of djay, the method by which you can install the software varies depending on where you originally purchased the app: the Mac App Store or the Algoriddim website.
Purchased in the Mac App Store:
Older versions of djay are no longer available for new purchases in the Mac App Store. This is why you will not find them via the Mac App Store search. You will still be able to download the version you purchased in the Purchased section of your account. Please see this detailed guide on how to redownload an application.
Purchased via the Algoriddim website:
Please use the following link to download the demo of the corresponding version and enter your activation key when prompted:
Note: Only the Algoriddim website version requires an activation key. If you're asked for an activation key, but you're sure you purchased djay from the Mac App Store, please see this FAQ.