This happens because Mac OS X doesn’t load the Bluetooth Low Energy driver at the login stage. Your K580 uses Bluetooth Low Energy 4.1 and won't be recognized when you first turn on your Mac. 
NOTE: You won't encounter this issue when you wake the device from sleep mode.

Follow either of the following suggestions to log in:

• Type your password using the integrated keyboard on your laptop — the Bluetooth connection will be restored after you log in.
• Connect your keyboard to your Mac laptop or desktop using the included USB receiver.