I know that it is is not possible to get the App download numbers from any API except by using paid service however I found out that there is a Formula which finds number of downloads on daily basis.
number of iPhone downloads = 52,958 * rank-(0.944)
number of iPad downloads = 13,516 * rank(-0.903)
However I found out that this is merely an approximated value and a daily value, so do we have to multiply it with the number of days since the day the application is submitted? It does not seem to make a lot of sense to me.
Will the above solution work to find an approximation of the amount of downloads of an app?
Is there any API to definitely find the rank instead of having a daily one?
I'm guessing the formula you found should look more like this:
52958 * (rank^-0.944)
That is, rank raised to the power of -0.944, times around 52958. What this would be saying is that rank 1 on that day would have 52958 downloads. They would drop off exponentially:
rank 10: 6024
rank 100: 685
rank 1000: 77
I don't know how accurate this is. As you point out, the big number (which is rank 1's daily downloads) is going to vary. If you don't have a way to get this actual number, then the formula is just a guess multiplied by an approximation.
The behavior of the equation is right though: a few high-ranked apps sell a lot of copies but the vast majority of apps sell very little.