I could be wrong, but is it not two different platforms, therefore two different purchases required? One for OSX and one for IOS?
Check Settimgs>iTunes & App Stores>Apple ID. Is there the account that you use to buy this app?
It's not showing in itunes but it shows in applications on my computer.
You mean in Applications folder or in Apps Library in iTunes? If isn't in Apps Library drag it there from the folder. Then sync with iPhone.