# Floor Summation Closed Form?

Let $$a_1,a_2,a_3,...a_n$$ be a set of positive integers. Does there exist any closed form for the approximation of the sum $$\sum_{i=p}^n\operatorname{floor}\left(\frac{a_p}{i-p+1}\right) ?$$

If not so, can we atleast get a recursive relation between $$sum(p)$$ and $$sum(p-1)$$? Call $$sum(p) = \sum_{i=p}^n\operatorname{floor}\left(\frac{a_p}{i-p+1}\right)$$

The relation should be like if I have the value of $$sum(p)$$ and I know what $$a_{p-1}$$ is, I can get the value of $$sum(p-1)$$. Can we get a recursive relation like this?