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
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:
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 how many coordinate pairs there are such that
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.