As far as I can remember, once you have downloaded the Kindle app and linked it to your Amazon account, when you purchase a Kindle book you have the option to automatically send it to your registered Kindle app via your Amazon account settings (via 'Your content and devices' IIRC). I certainly see this behaviour on my iPad and I'm pretty sure this was the case on my Nexus when i was using it ay back when (I can't test this as it needs charging).
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