Let $$f$$ be a polynomial function such that $$\log_2(f(x)) = \left\lfloor \log_2\left(2 + \frac{2}{3} + \frac{2}{9} + \ldots \infty\right) \right\rfloor \cdot \log_3\left(1 + \frac{f(x)}{f\left(\frac{1}{x}\right)}\right)$$, $$x > 0$$ and $$f(6) = 37$$. Then $$\sum_{n=1}^{10} f(n)$$ is equal to __________.

$$f(x) \cdot f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right) - 1$$

$$f(x) = x^n + 1 \quad \text{or} \quad f(x) = -x^n + 1$$

$$S = 2 + \frac{2}{3} + \frac{2}{9} + \dots \infty$$

$$S = \frac{2}{1 - \frac{1}{3}} = \frac{2}{\frac{2}{3}} = 3$$

$$\lfloor \log_2(S) \rfloor = \lfloor \log_2(3) \rfloor$$

$$\lfloor \log_2(3) \rfloor = 1$$

$$\log_2(f(x)) = 1 \cdot \log_3\left(1 + \frac{f(x)}{f\left(\frac{1}{x}\right)}\right)$$

$$\log_2(f(x)) = \log_3\left(\frac{f\left(\frac{1}{x}\right) + f(x)}{f\left(\frac{1}{x}\right)}\right)$$

$$f(x) = 2^k \quad \text{and} \quad \frac{f(x) + f\left(\frac{1}{x}\right)}{f\left(\frac{1}{x}\right)} = 3^k$$

$$f(x) = \pm x^n + 1$$

$$f(6) = 6^n + 1 = 37$$

$$6^n = 36 \implies n = 2$$

$$f(x) = x^2 + 1$$

$$\sum_{n=1}^{10} f(n) = \sum_{n=1}^{10} (n^2 + 1) = \sum_{n=1}^{10} n^2 + \sum_{n=1}^{10} 1$$

$$\sum_{n=1}^{10} n^2 = \frac{10 \times 11 \times 21}{6} = 385$$

$$\sum_{n=1}^{10} 1 = 10 \times 1 = 10$$

$$\text{Total Sum} = 385 + 10 = 395$$