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Thread: Closed form for Partial Sum of a Floor Function

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    Closed form for Partial Sum of a Floor Function

    Hello, I was wondering if there is a way to calculate a closed form for the partial sum of a floor function. By closed form, I mean something analog to \sum\limits_{k=1}^{n} k = \frac{n(n+1)}{2}
    ex.

    \sum\limits_{k=1}^{n} \lfloor\frac{60}{k}\rfloor = ?

    Motivation
    Started with the simple problem that if I have an alarm clock that displays hours and minutes, how many combinations are possible such that the number of minutes is a non-zero multiple of the hour value. ex. 12:48 is one such time as 48 is a non-zero multiple of 12.

    Found that my sum would be:
    \sum\limits_{k=1}^{12} \lfloor\frac{60}{k}\rfloor minus the cases where k evenly divides 60 per term. (as to exclude the invalid case of 12:60, etc.)

    Wanted to know if there was a way to calculate, without loss of generality, given integers k \leq K, l, M how many coordinate pairs (k,lk) there are such that lk < M, lk \neq 0
    or would you say this would simply be done by brute force using a computer algorithm?

    Also, if there are general theorems or resources to look at for summation for floor/ceiling functions, this will also be acceptable as a response.
    Last edited by MacstersUndead; Jan 17th 2017 at 11:35 AM.
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