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I am trying to find the sum of the following series asked by my friend. $$n\cdot\left(\bigl\lfloor\tfrac{n}{2}\bigr\rfloor+ \bigl\lfloor\tfrac{n}{3}\bigr\rfloor+ \bigl\lfloor\tfrac{n}{4}\bigr\rfloor+ \bigl\lfloor\tfrac{n}{5}\bigr\rfloor+ \cdots\right)$$ where the sum is up to $n/2$ terms if $n$ is even, and up to $(n-1)/2$ terms if $n$ is odd.

I am not getting how to solve it. Is this form can be reduced to a formula? Thank you.

Note: $\bigl\lfloor\;\;\bigr\rfloor$ means floor function, like $\bigl\lfloor 4.78\bigr\rfloor=4$.

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  • $\begingroup$ What does 《...》 mean? $\endgroup$ – Zev Chonoles Sep 8 '13 at 17:07
  • $\begingroup$ If $n$ is odd, what do you mean by "$n/2$ terms"? $\endgroup$ – Zev Chonoles Sep 8 '13 at 17:09
  • $\begingroup$ M realy very sorry... plz have a look on updated quedtion $\endgroup$ – avaneesh kumar Sep 8 '13 at 17:17

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