Consider the following sequence of operations :
Start with a natural number $n$ and then round it up to the closest multiple of $n-1$ .Then round up this new number to the closest multiple of $n-2$ and so on until you round to a multiple of $1$ . Call this last number $f(n)$ .
For example for $n=11$ the sequence is : $11,20,27,32,35,36,40,40,42,42,42$ and this means that $f(11)=42$
At first it seemed to me that this sequence is extremely hard to predict because we can't know how the multiples are distributed .
But to my amazement the following limit holds :
Prove that: $$\lim_{n \to \infty} \frac{n^2}{f(n)}=\pi$$
What I found :
- If $k \leq \lfloor \frac{n}{2} \rfloor$ then after the $k$-th operation the number is $(k+1)(n-k)$ .
This is not hard to prove by induction on $k$ . As an example the sequence for $n=11$ can be seen as : $$11,2 \cdot 10,3 \cdot 9,4 \cdot 8 ,5 \cdot 7 ,6 \cdot 6, \ldots $$
But after this we have $40=5 \cdot 8$ that breaks the pattern .I couldn't find any such patterns for $k>\lfloor \frac{n}{2} \rfloor$ and the sequence afterwards seems pretty arbitrary .
I am still amazed by how $\pi$ plays a role in this pure number theoretic question and I'd love to see a proof of the limit (or at least something new about the sequence ).
Thanks for everyone that can help me with this extraordinary problem .