to clarify the difference between this and the supposed duplicate, these two questions talk about completely different functions with completely different purposes
I was given this from a friend. They asked me to deduce what the equation is of. I played around with trying to compute alpha for some time. Plugging it into f(x), the function appeared to equal $0$ almost everywhere. I could never find the actual value of alpha. It appears to be an infinite irrational decimal number starting with $1.1973...$.
Let $$\alpha = \sum_{m=1}^{\infty} m * (2^{-(\sum_{j=2}^{m} (\lfloor \frac {(j-1)! + 1}{j} \rfloor - \lfloor \frac {(j-1)!}{j} \rfloor))^2}) \\* \left(\left\lfloor \frac {\sum_{i = 1}^{\lfloor \sqrt{m} \rfloor + 1} (\lfloor \frac {m}{i} \rfloor - \lfloor \frac {m-1}{i} \rfloor)-1}{\sqrt {(\sum_{i = 1}^{\lfloor \sqrt{m} \rfloor + 1} (\lfloor \frac {m}{i} \rfloor - \lfloor \frac {m-1}{i} \rfloor)-1)^2 + 1}} \right\rfloor + 1\right) * \left(\left\lfloor \frac {-\sum_{i = 1}^{\lfloor \sqrt{m} \rfloor + 1} (\lfloor \frac {m}{i} \rfloor - \lfloor \frac {m-1}{i} \rfloor)+1}{\sqrt {(\sum_{i = 1}^{\lfloor \sqrt{m} \rfloor + 1} (\lfloor \frac {m}{i} \rfloor - \lfloor \frac {m-1}{i} \rfloor)-1)^2 + 1}} \right\rfloor + 1\right)$$
Then let $f(x) = \lfloor 2^{x^2}* \alpha \rfloor - 2^{2x-1}*\lfloor 2^{(x-1)^2}*\alpha \rfloor$.
What is the function f(x)? Is there any way to reduce alpha to something simpler? I think this equation isn't something trivial. It doesn't appear to be $0$. It seems bizarre that alpha consists of so many sums. How do I reduce them into something simpler?