This is a part of my research about irrationality measure of $pi$ and sum of power divisor function $sigma$, I have used the following ratio of odd iteration and even iteration of sum power divisor function $dfrac{sigma_{2n+1}(p)}{sigma_{2n+2}(p)}$ for $p=7$ which the ratio close to $1/7$. Multiply $dfrac{sigma_{2n+1}(p)}{sigma_{2n+2}(p)}$ by $22$ the value of that sequences would be close to $22/7$. It is known that $sigma_n(p)=n^p+1$ with $p$ is a prime number. Just a small improvement I come up to the following integer sequences $ a_n $, and $ b_n $ such that :

$$

a_n=22*(4times7^{2n+1}-4times^{2n-3}+7^{2n-4})

quadtext{and}quad

b_n=4times7^{2n+2},

quad n>0

$$

such that $ a_n/b_n to pi$ with approximation of $10 ^{-6}$. Now my research is the determination of the values of both $C$ and $delta$ to get a bound for the difference in the below absolute value or LHS of the following inequality:

begin{equation}

leftlvertpi-22dfrac{4times7^{2n+1}-4times7^{2n-3}+7^{2n-4}}{4times7^{2n+2}}rightrvert<dfrac{C}{(4times7^{2n+2})^{1+delta}}.tag{1}label{1}

end{equation}

With the choice of $C=e^{e^{e^{e^{e}}}}$ the inequality holds up to $10^{10}$ using mathematica code with $delta$ in the range $(0,1)$. Now I have two question regarding the prediction of irrationality measure of $pi$ rathar than that the upper bound of LHS of inequality eqref{1}:

Question

- Must $delta$ always lie in $(0,1)$ with the choice of large real number $C$ and with the choice of integer sequences $ a_n $ and $ b_n $?
- Is it possible to find integer polynomial $P_n$ to make the ratio $dfrac{a_n+p_n}{b_n+p_n}$ as close as possible to $pi$ which lead to have a stronger bound which it self lead to predict the exact irrationality measure of $pi$?