Solution:
First, we factor
$$\begin{aligned} 27x^{6} + 1 &= (3x^{2})^{3} + 1 \\ &= (3x^{2} + 1)(9x^{4} - 3x^{2} + 1) \\ &= (3x^{2} + 1)((9x^{4} + 6x^{2} + 1) - 9x^{2}) \\ &= (3x^{2} + 1)((3x^{2} + 1)^{2} - (3x)^{2}) \\ &= (3x^{2} + 1)(3x^{2} + 3x + 1)(3x^{2} - 3x + 1) \end{aligned}$$
Letting $x = 10$, we get that $27,000,001 = 301 \cdot 331 \cdot 271$. A quick check shows that $301 = 7 \cdot 43$, so that $27,000,001 = 7 \cdot 43 \cdot 271 \cdot 331$. Each factor here is prime, and their sum is $652$.