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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?

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