First things first, are you sure it’s a Particle Core, and not a Particle Photon? Make sure you’re using the correct app. If it’s indeed a Core, you could try a factory reset, and then try again. With a Photon, there are some other steps which we can take if it’s indeed a Photon.
When you say “breathing blue”, is it purely blue, or could it be Cyan? If it’s cyan, it means it’s connected to the cloud. At this point, you should figure out the device ID. You can do this with the Particle CLI, but a regular serial terminal should suffice as well. In both cases, place it in listening mode, by holding the setup/mode button until it starts blinking blue. Then, using the CLI do
particle identify or connect to it and press
i if you’re using a serial terminal (putty for example).
Go to the web IDE and open the devices tab, there, click “add new device”. Enter the device ID you got.
After that, the device should show up on your account.