Playing around with numbers I stumbled upon the sequence that begins
$1,3,7,13,19,27...$
Looking it up on OEIS it is $A000960$ and is also known as the Flavius Josephus sequence. It is generated by taking all the positive integers and "sieving" out every other number, the remaining list has every third number sieved out, after that the remaining list has every fourth number sieved out, and so on ad infinitum.
$1,2,3,4,5,6,7,8,9,10,11,12,13...$
$1,3,5,7,9,11,13,...$
$1,3,7,9,13,...$
$1,3,7,13,...$
$...$
This sequence has several interesting properties, but what caught my eye was a comment left on OEIS claiming that for any number $n$, the amount of the number of terms in the sequence less than or equal to n is $2\sqrt{\frac{n}{\pi}}+O(n^\frac{1}{6})$ which would mean that the sequence grows like $\frac{\pi n^2}{4}$
There is a paper linked in the OEIS page explaining this, I am having trouble interpreting it, mainly due to the fact that it is in German.
I would appreciate any insight into why this particular sequence is connected to $\pi$
OEIS Link:https://oeis.org/A000960
German Paper:http://matwbn.icm.edu.pl/ksiazki/aa/aa85/aa8542.pdf
For the fun of it, I generated the values of the sequence up to $n=2000$ and made a quick and dirty curve fit to a power law $y=a n^b$. The results are $$\begin{array}{clclclclc} \text{} & \text{Estimate} & \text{Standard Error} & \text{Confidence Interval} \\ a & 0.78487 & 0.00029 & \{0.78430,0.78543\} \\ b & 2.00009 & 0.00005 & \{1.99999,2.00019\} \\ \end{array}$$ while $\frac \pi 4 \approx 0.78540$.
To give less weight to large number, using $$\log(y)=\log(a)+b\log(n)$$ we have $$\begin{array}{clclclclc} \text{} & \text{Estimate} & \text{Standard Error} & \text{Confidence Interval} \\ a & 0.78584 & 0.00078 & \{0.78431,0.78737\} \\ b & 1.99990 & 0.00015 & \{1.99960,2.00019\} \\ \end{array}$$
Now, for more fun, let $n=2\times 10^k$; the values are $$\left( \begin{array}{cc} k & \text{value} \\ 0 & 3 \\ 1 & 307 \\ 2 & 31483 \\ 3 & 3141667 \\ 4 & 314159547 \\ 5 & 31416163627 \\ 6 & 3141592399383 \\ 7 & 314159276364807 \end{array} \right)$$ Quite appealing, isn't it ?