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I was watching this video from James Grime on his channel singingbanana: https://www.youtube.com/watch?v=LejoPGtliTs and I wondered how many times does he have to flip every othelo piece.

So I found this sum: $$\sum_{k=1}^n \left \lfloor \frac{n}{k} \right \rfloor$$

Is there a way to compute it with a formula?

Simply Beautiful Art
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pavle
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    This defines sequence A006218 in OEIS. I don't imagine there is a simple expression for it, but on the OEIS page you can find more detailed discussion. – lulu Oct 16 '17 at 16:47

1 Answers1

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Partial answer:

The Othello piece with label $k$ gets flipped once for each factor of $k$. For instance $10$ gets flipped when every piece gets flipped (factor $1$), when every second piece gets flipped (factor $2$), when every fifth piece gets flipped (factor $5$), and when every tenth piece gets flipped (factor $10$).

So the total number of flips is $\tau(1)+\tau(2)+\cdots+\tau(90)$, where $\tau(k)$ is the divisor counting function--it returns the number of positive integer divisors (factors) of $k$. But this may be unsatisfying as it trades a sum for a sum.

There is considerable information on the sum ${\sum}_{k=1}^n \tau(k)$ on the internet. For example: http://www.maths.lancs.ac.uk/jameson/cdiv.pdf

paw88789
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