Is there any way I can defend against automated file download bots?

My company offers a very specifically tailored Android application to supplement our other software. Since users must have an account in our other software in order to use the Android application, is not useful to anyone who does not already subscribe to use our other software. Because of this, we are in a position to know exactly how many Android app users we should have.

We host the application file (.apk file) on our own website and direct our first-time users to download the application from there. After that, we have an automated update system built into the app that notifies the users that there is an update available and it and will update their app for them if they choose to accept the update.

We have very basic analytics in place on our website to monitor manual vs. automatic (update) downloads of our APK. We can see what file was downloaded at what time and by what IP address. After several months of manual and automated download analytics numbers that match up with our user count, we suddenly logged several hundred more manual downloads than we have users last month.

The download pattern I observed when investigating is that the same IP address is downloading the APK in bursts of 2-9 times all within the span of about a minute, and then within another minute or two another IP address does the same thing. This happened on and off for several days and I suspect there is some kind of bot/automated software that found our APK and is now downloading several copies of it for reasons I can’t currently comprehend.

I am hoping to find out if there exists some server configuration, 3rd-party technology, or even some kind of website programming technique we may be able to put into place to protect our site from this behavior? I don’t have reason to believe this "bot" is causing us any monetary or intellectual harm at this point in time, but if this continues it will certainly render our download analytics useless.