AirPods disconnect after iPhone 16 Pro Max upgrade

Re: "Recently upgraded from a 13 pro max to the 16 pro max. ....."

" ..... I’ve noticed my AirPods shut off and turn back on multiple times while I’m using them"

Wondering if the AirPods need a firmware update to remain compatible with the new iPhone:

See: About firmware updates for AirPods - Apple Support

If so: Tip: Leave AirPods connected to power overnight with the iPhone connected to wi-fi close by,

so they can update automatically,

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Else: If firmware is already up-to-date:

If your AirPods won't stay connected wirelessly, there might be interference with other wireless devices.

See how to: Resolve Wi-Fi and Bluetooth issues caused by wireless interference - Apple Support

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Re: I’ve also noticed my phone shows “iPhone 13 pro max” in my list of devices but that phone has been unpaired from my Bluetooth.

Wondering which list of devices it is still showing up in.

For Bluetooth device list:

Settings > Bluetooth, beside the device, tap the info button (i) 

then Forget this Device.

This will remove the accessory from the list of available Bluetooth devices.

To remove the iPhone 13 pro max from your list of Apple devices:

You can find out how to see and manage your trusted devices:

Check your Apple Account device list to find where you’re signed in - Apple Support

Eg says:

" Permanently remove a device from your account page

If you’re still signed in to your Apple Account on the device, it might reappear on your account page the next time your device connects to the internet. To permanently remove the device from your account page, sign out of iCloud, Media & Purchases, iMessage, FaceTime, and Game Center on that device, or erase that device."

For iCloud if needed: See how to: Remove a device from Find Devices on iCloud.com - Apple Support