The sum of a natural number and its reciprocal is

Let the natural number be x.

According to the given condition,

$x+\frac{1}{x}=\frac{65}{8}$

$\Rightarrow \frac{x^{2}+1}{x}=\frac{65}{8}$

$\Rightarrow 8 x^{2}+8=65 x$

$\Rightarrow 8 x^{2}-65 x+8=0$

$\Rightarrow 8 x^{2}-64 x-x+8=0$

$\Rightarrow 8 x(x-8)-1(x-8)=0$

$\Rightarrow(x-8)(8 x-1)=0$

$\Rightarrow x-8=0$ or $8 x-1=0$

$\Rightarrow x=8$ or $x=\frac{1}{8}$

∴ x = 8         (x is a natural number)

Hence, the required number is 8.