If the app was downloaded first to the phone, it will automatically download to your home computer when you sync. Provided your home computer is the one you normally sync to.
Your work set-up has me confused and I'm wondering if, in reality, you are synced to your work computer which is why you get the 'erase all' message when you try and sync at home. Or have I misunderstood you?
To download the app to your home computer, launch iTunes, go to the store, search for the app, then download. There are two ways to get apps on the iPhone, either directly on the phone or via your iTunes on your synced computer.
I'm very confused too. Something does not seem right.
The computer at work is new (2 months ago). I don't see how it can be synced with my phone, as all the apps I've ever owned (over the the last 12 months) are still in place on my phone. If I had re-synced my iphone (even by mistake) to my new work computer, surely this would have wiped the previous applications?
So, it's definitely not synced with the work machine.
However, it doesn't appear to be synced with my home machine either, as per my previous description. The weird thing is that all of my current apps, except the newest one (the one downloaded at work) are all visible in iTunes, but grayed out as a result of the "sync applications" tick box not being checked.
Also, when I launch iTunes and go to the store, I can't simply download the app... all I see is a "buy" button. And, before you ask, yes, I am logged in.
OK, so, I think there is something not quite right here, so I have a different question. How does one go about re-installing the Mac OS (computer not phone) without loosing all iPhone apps?
Thanks.