1. **Stating the problem:** We are given a function $$K = n \ln n + \ln \ln n - n + 2.25 \left(\frac{\ln n!}{\ln n}\right) + \frac{n^s}{\ln n}$$ and several values of $n$ with corresponding $K_n$ values. 2. **Understanding the formula:** - $\ln$ denotes the natural logarithm. - $n!$ is the factorial of $n$. - The term $\frac{\ln n!}{\ln n}$ involves the logarithm of factorial divided by logarithm of $n$. - $s$ is an exponent in $n^s$, but its value is not given explicitly. 3. **Important notes:** - Factorials grow very fast, so $\ln n!$ can be approximated using Stirling's formula: $$\ln n! \approx n \ln n - n + \frac{1}{2} \ln(2 \pi n)$$ - The function $K$ seems to be related to prime counting or number theory due to the values given. 4. **Intermediate work:** - For $n=100$, $K_{100} = 541$. - For $n=101$, $K_{101} = 547$. - For $n=149$, $K_{149} = 859$. - For $n=177$, $K_{177} = 1051$. - For $n=233$, $K_{233} = 1427$. - For $n=303$, $K_{303} = 1999$. - For $n=444$, $K_{444} = 3119$. - For $n=579$, $K_{579} = 4211$. - For $n=641$, $K_{641} = 4759$. - For $n=777$, $K_{777} = 5903$. - For $n=778$, $K_{778} = 5923$. - For $n=839$, $K_{839} = 6469$. - For $n=953$, $K_{953} = 7523$. - For $n=1000$, $K_{1000} = 7919$. 5. **Explanation:** - The function $K$ is evaluated at various $n$ values. - The values $K_n$ correspond closely to the prime counting function $\pi(n)$, which counts the number of primes less than or equal to $n$. - For example, $\pi(100) = 25$, but $K_{100} = 541$ is much larger, so $K$ is not $\pi(n)$ but may be related to prime sums or other number theory functions. 6. **Summary:** - The problem provides a complex formula and data points. - Without the value of $s$ or further instructions, we cannot simplify or solve for $K$ explicitly. - The data suggests $K$ grows roughly like the prime counting function times some factor. **Final answer:** The function $K$ is defined as $$K = n \ln n + \ln \ln n - n + 2.25 \left(\frac{\ln n!}{\ln n}\right) + \frac{n^s}{\ln n}$$ with given values for $n$ and $K_n$. Further information about $s$ or the purpose of $K$ is needed for explicit evaluation or simplification.